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Topology/Metric Spaces

Before we begin
Before we discuss topological spaces in their full generality, we will first turn our attention to a special type of topological space, a metric space. This abstraction has a huge and useful family of special cases, and it therefore deserves special attention. Also, the abstraction is picturesque and accessible; it will subsequently lead us to the full abstraction of a topological space.

Definition
A metric space is a Cartesian pair $$(X,d)$$ where $$X$$ is a non-empty set and $$d:X\times X\to[0,\infty)$$, is a function which is called the metric which satisfies the requirement that for all $$a,b,c\in X$$: Note that some authors do not require metric spaces to be non-empty. We annotate $$(X,d)$$ when we talk of a metric space $$X$$ with the metric $$d$$.
 * 1) $$d(a,b)=0$$ if and only if $$a = b$$
 * 2) $$d(a,b)=d(b,a)$$ (symmetry)
 * 3) $$d(a,c)\le d(a,b)+d(b,c)$$ (triangle inequality)

Examples

 * An important example is the discrete metric. It may be defined on any non-empty set X as follows
 * $$d(x,y)=\begin{cases}1&:x\ne y\\0&:x=y\end{cases}$$

 * On the set of real numbers $$\R$$, define $$d(x,y)=|x-y|$$ (The absolute distance between $$x$$ and $$y$$).  To prove that this is indeed a metric space, we must show that $$d$$ is really a metric. To begin with, $$d(x,y)=|x-y|\ge0$$ for any real numbers $$x$$ and $$y$$.
 * $$d(x,y)=|x-y|=0\iff(x-y)=0\lor-(x-y)=0\iff x=y$$
 * $$d(x,y)=|x-y|=|y-x|=d(y,x)$$
 * $$d(x,z)=|x-z|=|x-y+y-z|=\bigr|(x-y)+(y-z)\bigr|\le|x-y|+|y-z|=d(x,y)+d(y,z)$$
 * On the plane $$\R^2$$ as the space, and let $$d\bigl((x_1,y_1),(x_2,y_2)\bigr)=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$.
 * This is the euclidean distance between $$(x_1,y_1)$$ and $$(x_2,y_2)$$).

 * We can generalize the two preceding examples. Let $$V$$ be a normed vector space (over $$\R$$ or $$\Complex$$). We can define the metric to be: $$d(x,y)=\|x-y\|$$. Thus every normed vector space is a metric space.
 * For the vector space $$\Complex^n$$ we have an interesting norm. Let $$x=(x_1,\ldots,x_n)$$ and $$y=(y_1,\ldots,y_n)$$ two vectors of $$\Complex^n$$. We define the p-norm: $$\|x\|_p=\left(\sum_{i=1}^n|x_i|^p\right)^\frac1p$$. For each $$p$$-norm there is a metric based on it. Interesting cases of $$p$$ are:
 * $$p=1$$. The metric is $$d(x,y)=\|x-y\|_1=\sum_{i=1}^n|x_i-y_i|$$
 * $$p=2$$. The metric is good-old Euclid metric $$d(x,y)=\|x-y\|_2=\sqrt{\sum_{i=1}^n|x_i-y_i|^2}$$
 * $$p=\infty$$. This is a bit surprising: $$d(x,y)=\|x-y\|_\infty=\max_{i=1\ldots n}\bigl\{|x_i-y_i|\bigr\}$$ As an exercise, you can prove that $$\lim_{p\to\infty}\|x\|_p=\max_{i=1\ldots n}\{|x_i|\}$$ thus justifying the definition of $$\|\cdot\|_\infty$$.
 * The great-circle distance between two points on a sphere is a metric.
 * The Hilbert space is a metric space on the space of infinite sequences $$\{a_k\}$$ such that $$\sum_{i=1}^\infty a_k^2$$ converges, with a metric $$d\bigl(\{a_i\},\{b_i\}\bigr)=\sqrt{\sum_{i=1}^\infty(a_i-b_i)^2}$$.
 * The concept of the Erdős number suggests a metric on the set of all mathematicians. Take $$x,y$$ to be two mathematicians, and define $$d(x,y)$$ as 0 if $$x,y$$ are the same person; 1 if $$x,y$$ have co-authored a paper; $$n$$ if the shortest sequence $$(\{x,a_1\},\{a_1,a_2\},\ldots,\{a_{n-1},y\})$$, where each step pairs two people who have co-authored a paper, is of length $$n$$; or $$\infty$$ if $$x\ne y$$ and no such sequence exists. This metric is easily generalized to any reflexive relation (or undirected graph, which is the same thing).  Note that if we instead defined $$d(x,y)$$ as the sum of the Erdős numbers of $$x,y$$, then $$d$$ would not be a metric, as it would not satisfy $$d(x,y)=0\iff x=y$$. For example, if $$x=y$$ = Stanisław Ulam, then $$d(x,y)=2$$.

Note
Throughout this chapter we will be referring to metric spaces. Every metric space comes with a metric function. Because of this, the metric function might not be mentioned explicitly. There are several reasons:
 * We don't want to make the text too blurry.
 * We don't have anything special to say about it.
 * The space has a "natural" metric. E.g. the "natural" metric for $$\R^n,\Complex^n$$ is the euclidean metric $$d_2$$.

As this is a wiki, if for some reason you think the metric is worth mentioning, you can alter the text if it seems unclear (if you are sure you know what you are doing) or report it in the talk page.

Motivation
The open ball is the building block of metric space topology. We shall define intuitive topological definitions through it (that will later be converted to the real topological definition), and convert (again, intuitively) calculus definitions of properties (like convergence and continuity) to their topological definition. We shall try to show how many of the definitions of metric spaces can be written also in the "language of open balls". Then we can instantly transform the definitions to topological definitions.

Definition
Given a metric space $$(X,d)$$ an open ball with radius $$r$$ around $$p$$ is defined as the set
 * $$B_r(p)=\bigl\{x\in X:d(x,p)<r\bigr\},(r\in\R^+)$$

Intuitively it is all the points in the space, that are less than $$r$$ distance from a certain point $$p$$.

Examples
Why is this called a ball? Let's look at the case of $$\R^3$$:
 * $$d\bigl((x_1,x_2,x_3),(y_1,y_2,y_3)\bigr)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2}$$

Therefore $$B_r\bigl((0,0,0)\bigr)$$ is exactly $$x_1^2+x_2^2+x_3^2<r^2$$ – The ball with $$(0,0,0)$$ at center, of radius $$r$$. In $$\R^3$$ the ball is called open, because it does not contain the sphere ($$x_1^2+x_2^2+x_3^2=r^2$$).

The Unit ball is a ball of radius 1. Lets view some examples of the $$B_1\bigl((0,0)\bigr)$$ unit ball of $$\R^2$$ with different $$p$$-norm induced metrics. The unit ball of $$\R^2$$ with the norm $$\|\cdot\|_p$$ is:
 * $$B_1\bigl((0,0)\bigr)=\Big\{(x,y)\in\R^2:d\bigl((x,y),(0,0)\bigr)<1\Big\}=\Big\{(x,y):\bigl\|(x,y)-(0,0)\bigr\|_p<1\Big\}=\Big\{(x,y):\|(x,y)\|_p<1\Big\}=\Big\{(x,y):\sqrt[p]{|x|^p+|y|^p}<1\Big\}$$

 * The metric induced by $$\|\cdot\|_1$$ in that case, the unit ball is: $$|x|+|y|<1$$
 * The metric induced by $$\|\cdot\|_2$$ in that case, the unit ball is: $$\sqrt{x^2+y^2}<1$$
 * The metric induced by $$\|\cdot\|_\infty$$ in that case, the unit ball is: $$\max\bigl\{|x|,|y|\bigr\}<1$$

As we have just seen, the unit ball does not have to look like a real ball. In fact sometimes the unit ball can be one dot:
 * The discrete metric, The unit ball is $$B_1\bigl((0,0)\bigr)=\Big\{d\bigl((0,0),(x,y)\bigr)<1\Big\}=\Big\{d\bigl((0,0),(x,y)\bigr)=0\Big\}=\{(0,0)\}$$

Definitions
Definition: We say that x is an interior point of A iff there is an $$\epsilon>0$$ such that: $$B_\epsilon(x)\sube A$$. This intuitively means, that x is really 'inside' A - because it is contained in a ball inside A - it is not near the boundary of A.

Illustration: Definition: The interior of a set A is the set of all the interior points of A. The interior of a set A is marked $$\operatorname{int}(A)$$. Useful notations: $$\operatorname{int}(S)=\{x\in S|x\text{ is an interior point of }S\}$$ and $$\operatorname{int}(S)=\cup \{A\subseteq S|A\text{ is open }\!\!\}\!\!\text{ }$$.

Properties
Some basic properties of int (For any sets A,B):
 * $$\operatorname{int}(A) \subseteq A$$
 * $$\operatorname{int}(\operatorname{int}(A)) = \operatorname{int}(A)\,$$
 * $$\operatorname{int}(A \cap B) = \operatorname{int}(A) \cap \operatorname{int}(B)$$
 * $$A \subseteq B \Rightarrow \operatorname{int}(A) \subseteq \operatorname{int}(B)$$

Proof of the first: We need to show that: $$ x\in \operatorname{int}(A) \implies x \in A$$. But that's easy! by definition, we have that $$x\in B_\epsilon(x)\subseteq A$$ and therefore $$x\in A$$

Proof of the second: In order to show that $$\operatorname{int}(\operatorname{int}(A)) = \operatorname{int}(A)\,$$, we need to show that  $$\operatorname{int}(\operatorname{int}(A)) \subseteq \operatorname{int}(A)$$ and  $$\operatorname{int}(\operatorname{int}(A)) \supseteq \operatorname{int}(A)$$.  The " $$\subseteq$$" direction is already proved: if for any set A, $$\operatorname{int}(A) \subseteq A$$, then by taking $$\operatorname{int}(A)$$ as the set in question, we get $$\operatorname{int}(\operatorname{int}(A)) \subseteq \operatorname{int}(A)$$. The " $$\supseteq$$" direction: let $$x \in \operatorname{int}(A)$$. We need to show that $$x \in \operatorname{int}(\operatorname{int}(A))$$.  If $$x\in \operatorname{int}(A)$$ then there is a ball $$B_\epsilon(x) \subseteq A$$. Now, every point y, in the ball $$B_\frac{\epsilon}{2}(x) $$ an internal point to A (inside $$\operatorname{int}(A)$$), because there is a ball around it, inside A: $$y \in B_\frac{\epsilon}{2}(x) \implies B_\frac{\epsilon}{2}(y) \subset B_\epsilon(x) \subset A$$. We have that $$x\in B_\frac{\epsilon}{2}(x) \subset \operatorname{int}(A)$$ (because every point in it is inside $$\operatorname{int}(A)$$) and by definition $$x \in \operatorname{int}(\operatorname{int}(A))$$.
 Hint: To understand better, draw to yourself $$x, B_\epsilon(x), B_\frac{\epsilon}{2}(x), y, B_\frac{\epsilon}{2}(y)$$.

Proof of the rest is left to the reader.

Reminder

 * [a, b] : all the points x, such that $$a \leq x \leq b $$
 * (a, b) : all the points x, such that $$a < x < b $$

Example
For the metric space $$\mathbb{R}$$ (the line), we have:
 * $$int([a,b]) = (a,b)$$
 * $$int((a,b]) = (a,b)$$
 * $$int([a,b)) = (a,b)$$
 * $$int((a,b)) = (a,b)$$

Let's prove the first example ($$int([a,b]) = (a,b)$$). Let $$x\in (a,b)$$ (that is: $$a < x < b $$) we'll show that $$x$$ is an internal point.
 Let $$\epsilon = \min\{x - a, b - x \}$$. Note that $$x + \epsilon \leq x + b - x = b$$ and $$x - \epsilon \geq x - x + a = a$$. Therefore $$ B_\epsilon(x) = (x - \epsilon, x + \epsilon) \subset (a,b)$$.
 We have shown now that every point x in $$(a,b)$$ is an internal point. Now what about the points $$a,b$$ ? let's show that they are not internal points. If $$a$$ was an internal point of $$[a,b]$$, there would be a ball $$B_\epsilon(a) \subset [a,b]$$. But that would mean, that the point $$ a - \frac{\epsilon}{2}$$ is inside $$[a, b]$$. but because $$ a - \frac{\epsilon}{2} < a $$ that is a contradiction. We show similarly that b is not an internal point.
 To conclude, the set $$(a,b)$$ contains all the internal points of $$[a,b]$$. And we can mark $$int([a,b]) = (a,b)$$

Definition
A set is said to be open in a metric space if it equals its interior ($$ A = Int(A)$$). When we encounter topological spaces, we will generalize this definition of open. However, this definition of open in metric spaces is the same as that as if we regard our metric space as a topological space.

Properties:
 * 1) The empty-set is an open set (by definition: $$int(\emptyset)=\emptyset$$).
 * 2) An open ball is an open set.
 * 3) For any set B, int(B) is an open set. This is easy to see because: int(int(B))=int(B).
 * 4) If A,B are open, then $$A\cap B$$ is open.  Hence finite intersections of open sets are open.
 * 5) If $${A_i: i \in I}$$ (for any set if indexes I) are open, then their union $$\cup_{i\in I} A_i$$ is open.

Proof of 2:
 Let $$B_r(x)$$ be an open ball. Let $$y \in B_r(x)$$. Then $$y \in B_{r-d(x,y)}(y) \subseteq B_r(x)$$.
 In the following drawing, the green line is $$d(x,y)$$ and the brown line is $$r-d(x,y)$$. We have found a ball to contain $$y$$ inside $$B_r(x)$$.

Proof of 4:
 A, B are open. we need to prove that $$int(A \cap B) = A\cap B$$. Because of the first propriety of int, we only need to show that $$int(A \cap B) \supseteq A\cap B$$, which means $$\forall x\in A\cap B:x\in int(A \cap B)$$. Let $$x \in A \cap B$$. We know also, that $$x \in int(A), x \in int(B)$$ from the premises A, B are open and $$x \in A, x \in B$$. That means that there are balls: $$B_{{\epsilon}_1}(x) \subset A, B_{{\epsilon}_2}(x) \subset B$$. Let $$\epsilon = \min\{{\epsilon_1, \epsilon_2}\}$$, we have that $$B_{\epsilon}(x) \subset A, B_{\epsilon}(x) \subset B \Rightarrow  B_{\epsilon}(x) \subset A\cap B $$. By the definition of an internal point we have that $$x\in int(A \cap B)$$ ($$B_{\epsilon}(x)$$ is the required ball).

Interestingly, this property does not hold necessarily for an infinite intersection of open sets. To see an example on the real line, let $$A_n=\{(-1/n,1/n)\}$$. We then see that $$\cap^\infty_{i=1}A_i=\{0\}$$ which is closed.

Proof of 5:
 Proving that the union of open sets is open, is rather trivial: let $${A_i: i \in I}$$ (for any set if indexes I) be a set of open sets. we need to prove that $$int(\cup_{i\in I} A_i) \supseteq \cup_{i\in I} A_i$$: If $$x\in A_i$$ then it has a ball $$B_\epsilon(x) \subset A_i \subseteq \cup_{i\in I} A_i$$. The same ball that made a point an internal point in $$A_i$$ will make it internal in $$\cup_{i\in I} A_i$$.

Proposition: A set is open, if and only if it is a union of open-balls.
 Proof: Let A be an open set. by definition, if $$x\in A$$ there there a ball $$B_{\epsilon_x}(x) \subseteq A$$. We can then compose A: $$A = \cup_{x\in A}B_{\epsilon_x}(x)$$. The equality is true because: $$ \cup_{x\in A}B_{\epsilon_x}(x) \subseteq A$$ because $$ \forall x \in A: B_{\epsilon_x}(x) \subseteq A$$. $$ \cup_{x\in A}B_{\epsilon_x}(x) \supseteq A$$ in each ball we have the element $$x$$ and we unite balls of all the elements of $$A$$.
 On the other hand, a union of open balls is an open set, because every union of open sets is open.

Examples
Proof: Let $$U$$ be a set. we need to show, that if $$x\in U$$ then $$x$$ is an internal point. Lets use the ball around $$x$$ with radius $$\frac{1}{2}$$. We have $$B_\frac{1}{2}(x) = \{y\mid d(x,y) < \frac{1}{2}\} = \{x\} \subseteq U$$. Therefore $$x$$ is an internal point.
 * As we have seen, every open ball is an open set.
 * For every space $$X$$ with the discrete metric, every set is open.
 * The space $$\mathbb{R}$$ with the regular metric. Every open segment $$(a,b)$$ is an open set. The proof of that is similar to the proof that $$int([a,b]) = (a,b)$$, that we have already seen.

Theorem
In any metric space X, the following three statements hold:
 * 1) The union of any number of open sets is open.
 * Proof: Let $$C$$ be a collection of open sets, and let
 * $$x \in \cup C$$. Then there exists a $$U \in C$$ such that $$x \in U$$.
 * So there exists an $$\epsilon>0$$ such that $$B_\epsilon(x) \subseteq U$$. Therefore
 * $$B_\epsilon(x) \subseteq \cup C$$.
 * 2) The intersection of a finite number of open sets is open.
 * Proof: Let $$x \in \cap C$$, where $$C$$ is a finite collection of open sets.
 * So $$x \in U$$ for each $$U \in C$$. Let $$C = {U_1,U_2,...,U_n}$$. For each $$i=1,2,3,...,n$$, there exists an $$\epsilon_i > 0$$ such that $$B_\epsilon(x) \subseteq U_i$$. Let $$\epsilon = min_i$${$$\epsilon_i$$}. Therefore $$ \epsilon>0$$ and $$B_\epsilon(x) \subseteq \cap C$$.
 * 3) The empty set and X are both open.

Theorem
In any metric space X, the following statements hold:
 * 1) The intersection of any number of closed sets is closed.
 * 2) The union of a finite number of closed sets is closed.

Definition
First, Lets translate the calculus definition of convergence, to the "language" of metric spaces: We say that a sequence $$x_n$$ converges to $$x$$ if for every $$\epsilon > 0$$ exists $$N$$ that for each $$ n^* > N$$ the following holds: $$d(x_{n^*},x) < \epsilon$$.
 Equivalently, we can define converges using Open-balls: A sequence $$x_n$$ converges to $$x$$ If for every $$\epsilon > 0$$ exists $$N$$ that for each $$ n^* > N$$ the following holds: $$x_{n^*} \in B_\epsilon(x)$$.

The latter definition uses the "language" of open-balls, But we can do better - We can remove the $$\epsilon$$ from the definition of convergence, thus making the definition more topological. Let's define that $$x_n$$ converges to $$x$$ (and mark $$x_n \rightarrow x$$), if for every ball $$B$$ around $$x$$ , exists $$N_B$$ that for each $$ n^* > N_B$$ the following holds: $$x_{n^*} \in B(x)$$. $$x$$ is called the limit of the sequence.

The definitions are all the same, but the latter uses topological terms, and can be easily converted to a topological definition later.

Properties

 * If a sequence has a limit, it has only one limit. Proof Let a sequence $$x_n$$ have two limits, $$x\,$$ and $$x^\prime$$. If they are not the same, we must have $$0<d(x,x^\prime)$$. Let $$\epsilon$$ be smaller than this distance. Now for some $$N$$, for all $$n>N$$, it must be the case that both $$x_n \in B_{\epsilon / 2}(x)$$ and $$x_n \in B_{\epsilon / 2}(x^\prime)$$ by virtue of the fact $$x\,$$ and $$x^\prime$$ are limits. But this is impossible; the two balls are separate. Therefore the limits are coincident, that is, the sequence has only one limit.
 * If $$x_n \rightarrow x$$, then almost by definition we get that $$d(x_n, x) \rightarrow 0$$. ($$d(x_n, x)$$ Is the sequence of distances).

Examples
The space $$\mathbb{R}^k$$ For any p-norm induced metric, when $$p\geq 1$$. Let $$\vec{x_n} = (x_{n,1},x_{n,2},\cdots, x_{n,k})$$. and let $$\vec{x} = (x_{1},x_{2},\cdots, x_{k})$$.
 Then, $$\vec{x_n} \rightarrow \vec{x}$$ If and only if $$ \forall i, 1\leq i \leq k: x_{n,i} \rightarrow x_{i}$$.
 * In $$\mathbb{R}$$ with the natural metric, The series $$x_n = \frac{1}{n}$$ converges to $$0$$. And we note it as follows: $$\frac{1}{n}\rightarrow 0$$
 * Any space, with the discrete metric. A series $$x_n$$ converges, only if it is eventually constant. In other words: $$x_n\rightarrow x$$ If and only if, We can find $$N$$ that for each $$ n^* > N$$, $$x_{n^*} = x$$
 * An example you might already know:

Uniform Convergence
A sequence of functions $$\{ f_n \}$$ is said to be uniformly convergent on a set $$S$$ if for any $$\epsilon>0$$, there exists an $$N$$ such that when $$a$$ and $$b$$ are both greater than $$N$$, then $$d(f_a(x),f_b(x)) < \epsilon$$ for any $$x \in S$$.

Closure
Definition: The point $$p$$ is called a point of closure of a set $$A$$ if there exists a sequence $$a_n, \forall n, a_n \in A$$, such that $$a_n \rightarrow p$$.

In other words, the point $$p$$ is a point of closure of a set $$A$$ if there exists a sequence in $$A$$ that converges on $$p$$. Note that $$p$$ is not necessarily an element of the set $$A$$.

An equivalent definition using balls: The point $$p$$ is called a point of closure of a set $$A$$ if for every open ball $$B$$ containing $$p$$, we have $$B \cap A \neq \emptyset$$. In other words, every open ball containing $$p$$ contains at least one point in $$A$$ that is distinct from $$p$$.
 The proof is left as an exercise.

Intuitively, a point of closure is arbitrarily "close" to the set $$A$$. It is so close, that we can find a sequence in the set that converges to any point of closure of the set.

Example: Let A be the segment $$[0,1) \in \mathbb{R}$$, The point $$p = 1$$ is not in $$A$$, but it is a point of closure: Let $$a_n = 1 - \frac{1}{n}$$. $$a_n \in A$$ ($$n > 0$$, and therefore $$a_n = 1 - \frac{1}{n} < 1$$) and $$ a_n \rightarrow 1 $$ (that's because $$ \frac{1}{n} \rightarrow 0$$).

Definition: The closure of a set $$A \subseteq X$$ $$({X},d)$$, is the set of all points of closure. The closure of a set A is marked $$\bar{A}$$ or $$Cl(A)$$.

Note that $$A \subseteq \bar{A}$$. a quick proof: For every $$x \in A$$, Let $$(a_n = x)\forall n$$.

Examples
For the metric space $$\mathbb{R}$$ (the line), and let $$a,b \in \mathbb{R}$$ we have:
 * $$Cl([a,b]) = [a,b]$$
 * $$Cl((a,b]) = [a,b]$$
 * $$Cl([a,b)) = [a,b]$$
 * $$Cl((a,b)) = [a,b]$$

Closed set
Definition: A set $$A \subseteq X$$ is closed in $${X}\,$$ if $$A = Cl(A)$$.
 Meaning: A set is closed, if it contains all its point of closure.

An equivalent definition is: A set $$A \subseteq X$$ is closed in $${X}\,$$ If for every point $$p \in A$$, and for every Ball $$B, p \in B$$, then $$B \cap A \neq \emptyset$$.
 The proof of this definition comes directly from the former definition and the definition of convergence.

Properties
Some basic properties of Cl (For any sets $$A,B$$):

This union can therefore not be a closed subset of the real numbers.
 * $$A \subseteq Cl(A)$$
 * $$Cl(Cl(A))=Cl(A)$$
 * $$Cl(A \cup B) = Cl(A)\cup Cl(B)$$
 * $$A $$ is closed iff $$A = Cl(A)$$
 * While the above implies that the union of finitely many closed sets is also a closed set, the same does not necessarily hold true for the union of infinitely many closed sets. To see an example on the real line, let $$A_n=\{[-1+\frac{1}{n},1-\frac{1}{n}]\}$$. We see that $$\cup_{i=1}^\infty A_i=(-1,1)$$ fails to contain its points of closure, $$\pm 1.$$

The proofs are left to the reader as exercises. Hint for number 5: recall that $$Cl(A)=\cap \{A\subseteq S|S\text{ is closed }\!\!\}\!\!\text{ }$$.

Open vs Closed
That is, an open set approaches its boundary but does not include it; whereas a closed set includes every point it approaches. These two properties may seem mutually exclusive, but they are not:

 * In any metric space $$(X, d)$$, the set $$X$$ is both open and closed.

 * In any space with a discrete metric, every set is both open and closed.

 * In $$\R$$, under the regular metric, the only sets that are both open and closed are $$\R$$ and $$\emptyset$$. However, some sets are neither open nor closed.  For example, a half-open range like $$[0, 1)$$ is neither open nor closed.  As another example, the set of rationals is not open because an open ball around a rational number contains irrationals; and it is not closed because there are sequences of rational numbers that converge to irrational numbers (such as the various infinite series that converge to $$\pi$$).

Complementary set
A Reminder/Definition: Let $$A$$ be a set in the space $$X$$. We define the complement of $$A$$, $$A^c$$ to be $$X \setminus A$$.

A Quick example: let $$X = [0,1]; A = [0,\frac{1}{2}]$$. Then $$A^c = (\frac{1}{2},1]$$.

The plot continues...
A very important Proposition: Let $$A$$ be a set in the space $$(X,d)$$. Then, A is open iff $$A^c$$ is closed.
 Proof: ($$\Rightarrow$$) For the first part, we assume that A is an open set. We shall show that $$ A^c = Cl(A^c)$$. It is enough to show that $$Cl(A^c) \subseteq A^c$$ because of the properties of closure. Let $$p \in Cl(A^c)$$ (we will show that $$p \in A^c$$).
 for every ball $$B, p\in B$$ we have, by definition that (*)$$B \cap A^c \neq \emptyset$$. If the point is not in $$A^c$$ then $$p \in A$$. $$A$$ is open and therefore, there is a ball $$B$$, such that: $$p \in B \subseteq A$$, that means that $$B \cap A^c = \emptyset$$, contradicting (*).
 ($$\Leftarrow$$) On the other hand, Lets a assume that $$A^c$$ is closed, and show that $$A$$ is open. Let $$p$$ be a point in $$A$$ (we will show that $$p \in int(A)$$). If $$p$$ is not in $$int(A)$$ then for every ball $$B, p \in B$$ we have that $$B \nsubseteq A$$. That means that $$B \cap A^c \neq \emptyset$$. And by definition of closure point $$p$$ is a closure point of $$A^c$$ so we can say that $$p \in Cl(A^c)$$. $$A^c$$ is closed, and therefore $$p \in A^c = Cl(A^c)$$ That contradicts the assumption that $$p \in A$$

Note that, as mentioned earlier, a set can still be both open and closed!

On $$\mathbb{R}$$
The following is an important theorem characterizing open and closed sets on $$\mathbb{R}$$.

Theorem: An open set $$O$$ in $$\mathbb{R}$$ is the union of countably many disjoint open intervals.

Proof: Let $$x\in O$$. Let $$a=\sup\{t|t\notin O, t<x\}$$ and let $$b=\inf\{t|t\notin O, t>x\}$$. There exists an open ball $$(x - \epsilon, x + \epsilon)$$ such that $$(x - \epsilon, x + \epsilon) \subseteq O$$ because $$O$$ is open. Thus, a≤x-ε and b≥x+ε. Thus, x ∈(a,b). The set O contains all elements of (a,b) since if a number is greater than a, and less than x but is not within O, then a would not be the supremum of {t|t∉O, t<x}. Similarly, if there is a number is less than b and greater than x, but is not within O, then b would not be the infimum of {t|t∉O, t>x}. Thus, O also contains (a,x) and (x,b) and so O contains (a,b). If y≠x and y∈(a,b), then the interval constructed from this element as above would be the same. If yy} would also be less than a because there is a number between y and a which is not within O. Similarly if y>b, then sup{t|t∉O, t<y} would also be greater than b because there is a number between y and b which is not within O. Thus, all possible open intervals constructed from the above process are disjoint. The union of all such open intervals constructed from an element x is thus O, and so O is a union of disjoint open intervals. Because the rational numbers is dense in R, there is a rational number within each open interval, and since the rational numbers is countable, the open intervals themselves are also countable.

Examples of closed sets

 * 1) In any metric space, a singleton $$\{x\}$$ is closed. To see why, consider the open set, $$\{x\}^c$$. Let $$y \in \{x\}^c$$. Then $$y \neq x$$, so $$d(y,x) > 0$$. Let $$\epsilon = \frac{1}{2}d(y,x)$$. Then $$B_\epsilon(y) \subseteq \{x\}^c$$. So $$\{x\}^c$$ is open, and hence $$\{x\}$$ is closed.
 * 2) In any metric space, every finite set $$T = \{x_1,x_2,...,x_n\}$$ is closed. To see why, observe that $$T^c = \Big[\bigcup\{x_i\}\Big]^c = \bigcap\{x_i\}^c$$ is open, so $$T$$ is closed.
 * 3) Closed intervals [a,b] are closed.
 * 4) Cantor Set Consider the interval [0,1] and call it C0. Let A1 be equal {0, $$\tfrac{2}{3}$$} and let dn = $$(\tfrac{1}{3})^{n}$$. Let An+1 be equal to the set An∪{x|x=a+2dn, a∈An}. Let Cn be $$\textstyle \bigcup_{a\in A_n}$${[a,a+dn]}, which is the finite union of closed sets, and is thus closed. Then the intersection $$\textstyle \bigcap_{i=1}^\infty {C_i}$$ is called the Cantor set and is closed.

Exercises

 * 1) Prove that a point x has a sequence of points within X converging to x if and only if all balls containing x contain at least one element within X.
 * 2) In $$\mathbb{R}$$ the only sets that are both open and closed are the empty set, and the entire set. This is not the case when you look at $$\mathbb{Q}\subseteq\mathbb{R}$$. Give an example of a set which is both open and closed in $$\mathbb{Q}$$.
 * 3) Let $$A$$ be a set in the space $$x$$. Prove the following:
 * 4) $$Cl(A) = Int(A^c)^c$$
 * 5) $$Int(A) = Cl(A^c)^c$$

Definition
Let's recall the idea of continuity of functions. Continuity means, intuitively, that you can draw a function on a paper, without lifting your pen from it. Continuity is important in topology. But let's start in the beginning:

The classic delta-epsilon definition: Let $$(X,d),(Y,e)$$ be spaces. A function $$f : X \rightarrow Y $$ is continuous at a point $$x$$ if for all $$\epsilon_x > 0$$ there exists a $$\delta_{\epsilon_x} > 0$$ such that: for all $$x_1$$ such that $$d(x,x_1) < \delta_{\epsilon_x} $$, we have that $$e(f(x), f(x_1)) < \epsilon_x$$.

Let's rephrase the definition to use balls: A function $$f : X \rightarrow Y $$ is continuous at a point $$x$$ if for all $$\epsilon_x > 0$$ there exists $$\delta_{\epsilon_x} > 0$$ such that the following holds: for every $$x_1$$ such that $$x_1 \in B_{\delta_{\epsilon_x}} (x) $$ we have that $$ f(x_1) \in B_{\epsilon_{x}}(f(x))$$. Or more simply: $$ f(B_{\delta_{\epsilon_x}}(x)) \subseteq B_{\epsilon_{x}}(f(x))$$

Looks better already! But we can do more.

Definitions:
 * A function is continuous in a set S if it is continuous at every point in S.
 * A function is continuous if it is continuous in its entire domain.

Proposition: A function $$f : X \rightarrow Y $$ is continuous, by the definition above $$\Leftrightarrow$$ for every open set $$U$$ in $$Y$$, The inverse image of $$U$$, $$f^{-1}(U)$$, is open in $$X$$. That is, the inverse image of every open set in $$Y$$ is open in $$X$$.
 Note that $$f$$ does not have to be surjective or bijective for $$f^{-1}$$ to be well defined. The notation $$f^{-1}$$ simply means $$f^{-1}(U) = \{x \in X: f(x) \in U\}$$.

Proof: First, let's assume that a function $$f$$ is continuous by definition (The $$\Rightarrow$$ direction). We need to show that for every open set $$U$$, $$f^{-1}(U)$$ is open.

Let $$U\subseteq Y$$ be an open set. Let $$x \in f^{-1}(U)$$. $$f(x)$$ is in $$U$$ and because $$U$$ is open, we can find and $$\epsilon_x$$, such that $$B_{\epsilon_x}(f(x)) \subseteq U$$. Because f is continuous, for that $$\epsilon_x$$, we can find a $$\delta_{\epsilon_x} > 0$$ such that $$ f(B_{\delta_{\epsilon_x}}(x)) \subseteq B_{\epsilon_{x}}(f(x)) \subseteq U$$. that means that $$B_{\delta_{\epsilon_x}}(x) \subseteq f^{-1}(U)$$, and therefore, $$x$$ is an internal point. This is true for every $$x$$ - meaning that all the points in $$f^{-1}(U)$$ are internal points, and by definition, $$f^{-1}(U)$$ is open.

($$\Leftarrow$$)On the other hand, let's assume that for a function $$f$$ for every open set $$U \in Y$$, $$f^{-1}(U)$$ is open in $$X$$. We need to show that $$f$$ is continuous.

For every $$x\in X$$ and for every $$\epsilon_x > 0$$, The set $$B_{\epsilon_x}(f(x))$$ is open in $$Y$$. Therefore the set $$V = f^{-1}(B_{\epsilon_x}(f(x)))$$ is open in $$X$$. Note that $$x\in V$$. Because $$V$$ is open, that means that we can find a $$\delta_{\epsilon_x}$$ such that $$B_{\delta_{\epsilon_x}}(x) \subseteq V$$, and we have that $$ f(B_{\delta_{\epsilon_x}}(x)) \subseteq B_{\epsilon_{x}}(f(x))$$.

The last proof gave us an additional definition we will use for continuity for the rest of this book. The beauty of this new definition is that it only uses open-sets, and there for can be applied to spaces without a metric, so we now have two equivalent definitions which we can use for continuity.

Examples

 * Let $$f$$ be any function from any space $$(X,d)$$, to any space $$(Y,e)$$, were $$d$$ is the discrete metric. Then $$f$$ is continuous. Why? For every open set $$U$$, the set $$f^{-1}(U)$$ is open, because every set is open in a space with the discrete metric.
 * Let $$f:\mathbb{R}\rightarrow\mathbb{R}; f(x)=x$$ The identity function. $$f$$ is continuous: The source of every open set is itself, and therefore open.

Exercise

 * 1) Prove that a function $$f : X \rightarrow Y $$ is continuous $$\Leftrightarrow$$ for every closed set $$U$$ in $$Y$$, The inverse image of $$U$$, $$f^{-1}(U)$$, is closed in $$X$$.

Uniform Continuity
In a metric space X, function from X to a metric space Y is uniformly continuous if for all $$\epsilon$$, there exists a $$\delta$$ such that for all $$x_1,x_2\in X$$, $$d(x_1,x_2)<\delta$$ implies that $$d(f(x_1),f(x_2))<\epsilon$$.

Isometry
An isometry is a surjective mapping $$f: X \rightarrow Y$$, where $$(X, \delta )$$ and $$(Y , \rho )$$ are metric spaces and for all $$a, b \in X$$, $$\delta (a, b) = \rho (f(a), f(b))$$.

In this case, $$(X, \delta )$$ and $$(Y, \rho )$$ are said to be isometric.

Note that the injectivity of $$f$$ follows from the property of preserving distance:

 * $$f(a)=f(b)$$
 * $$\implies\rho(f(a),f(b))=0$$
 * $$\implies\delta(a,b)=0$$
 * $$\implies a=b$$

So an isometry is necessarily bijective.

Exercises

 * 1) Show that a set is a metric open set iff it is a (possibly infinite) union of open balls.
 * 2) Show that the discrete metric is in fact a metric.

title: Topology/Metric Spaces


text:
Odontomachus pseudobauri

Odontomachus pseudobauri is an extinct species of ant in the subfamily Ponerinae known from one possibly Miocene fossil found on Hispaniola. O. pseudobauri is one of two species in the ant genus Odontomachus to have been described from fossils found in Dominican amber and is one of a number of Odontomachus species found in the Greater Antilles.

History and classification
Odontomachus pseudobauri is known from a solitary fossil insect which, along with a coleopteran and a dipteran, is an inclusion in a transparent yellow chunk of Dominican amber purchased from an amber dealer in Basel, Switzerland. The amber was produced by the extinct Hymenaea protera, which formerly grew on Hispaniola, across northern South America, and up to southern Mexico. The specimen was collected from an undetermined amber mine in fossil-bearing rocks of the Cordillera Septentrional mountains of northern Dominican Republic. The amber dates from the Burdigalian stage of the Miocene, based on studying the associated fossil foraminifera, and may be as old as the Middle Eocene, based on the associated fossil coccoliths. This age range is due to the host rock being secondary deposits for the amber, and the Miocene as the age range is only the youngest that it might be.

At the time of description, the holotype specimen, number "BMNHP-II32", was preserved in the Natural History Museum, London amber collections in London, England. The holotype fossil was first studied by entomologist Maria L. De Andrade of the University of Basle with her 1994 type description of the new species being published in the journal ''Stuttgarter Beiträge zur Naturkunde. Serie B (Geologie und Paläontologie)''. The specific epithet pseudobauri is derived from the Greek Latin word pseudo which means "false", and bauri from the modern species Odontomachus bauri.

Based on the head structure, O. pseudobauri was suggested to be part of the O. haematodus species group, closely placed with the species O. bauri. The modern species is found in the Galapagos islands, tropical regions of South and Central America, and most of the West Indies except the Bahamas and Cuba. O. bauri is one of two modern species found on the island of Hispaniola, and based on the similarities to O. pseudobauri De Andrade suggested the latter species might have been ancestral to the modern species. When first described, O. pseudobauri was one of two Odontomachus species that had been described from fossils. It and Odontomachus spinifer were both described by De Andrade from Dominican amber in the same paper. A third species Odontomachus paleomyagra, the first compression fossil species, was described in 2014 from a worker found in Priabonian age lignite deposits of the Most Basin, Czech Republic.

Description
The O. pseudobauri worker is approximately 9.28 mm in length, and has a shining exoskeleton of orange along the body, reddish head and mandibles, and brownish tones on the gaster. The smooth exoskeleton has tiny punctuation found across the rear and sides of the head, in the front areas of the antennae depressions, and over the gaster. In contrast the frons back across the rear areas of the antennae sockets are striated, with the striae becoming finer towards the rear of the depressions. The head has a rectangular outline, being 1/3 longer than wide, with the rear margin of the head wider than the maximum width of the pronotum. The mandibles are 3/4 as long as the head is wide and the chewing margin of the right mandible having 14 teeth, while there are 13 teeth in the left. The teeth of both margins increase in size towards the tip, while the apex of each has three teeth, a preapical, intercalary, and an apical tooth. The antennae have notably long scapes that extend past the rear margin of the head capsule. The first funicular segments of the antennae are 3/4 the length of the second segment. The pronotum and propodeum each have convex upper profiles, with approximately 25 concentric striae circling the pronotum. There is a large backward curving spine formed from the upper domed surface of the petiole, being a little narrower than the width of the petiole, and a smaller process is positioned on the forward underside area of the petiole. The worker is incomplete, missing the legs and antenna of the left side plus tip sections of the right legs. The third through end segments of the gaster are shrunken and missing.

source: wikimedia

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Odontomachus pseudobauri


text:
Talk:Karel Komárek

Edit request
Hello, I am working on behalf of Mr. Komarek and, in compliance with WP:PAID and Conflict of Interest guidelines, would like to request some updates to the page. They should make the page more accurate/up to date, and I have also tried to neutralise some of the older language.

I’ve supplied sources where I can, but if you have any questions please do ask below or on my talk page. Thanks in advance and I’m looking forward to hearing the community’s thoughts! Panepani (talk) 23:40, 31 October 2021 (UTC)

 * ✅ - as the article had been edited since your request was made, I've rewritten some of the requested information to fit in a little more. StartGrammarTime (talk) 08:06, 30 April 2022 (UTC)

1. In the career section, can

 * “The Group subsequently acquired majority stakes in cybersecurity firm AEC, software developer KOMIX, tech company Seavus, and consulting company Stratiteq.”

please be added after the paragraph

 * “In recent years, KKCG has made large investments in information technologies, including the acquisition of majority stakes in such companies as AUTOCONT[12] and Cleverlance,[13] two biggest parts of its holding ARICOMA Group.[14]”

Suggested sources would be
 * https://forbes-cz.translate.goog/aricoma-group-kupuje-komix-ziska-vyznamny-podil-na-digitalizaci-statni-spravy/?_x_tr_sl=cs&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=nui,sc
 * https://www.globallegalchronicle.com/aricoma-groups-acquisition-of-seavus/
 * https://forbes-cz.translate.goog/aricoma-posiluje-ve-skandinavii-koupila-svedskou-technologickou-spolecnost-stratiteq/?_x_tr_sl=cs&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=nui,sc

2. In the philanthropy section, could

 * “He is Chairman of the Board of Directors of the Karel Komárek Proměny Foundation, an organization that provides support for the improvement of public spaces, and Chairman of the Board of Directors of the Karel Komárek Family Foundation. He was instrumental in founding the Dvořák Prague International Music Festival.[18]”

be changed to

 * “Komarek is Chairman of the Board of Directors of the Karel Komárek Family Foundation (KKFF), which he set up with his wife, Štěpánka. The Foundation supports cultural landmarks, such as the Dvořák Prague International Music Festival, the John F. Kennedy Center for Performing Arts in Washington, and the Václav Havel Library in Prague.[18]”

3. Finally, in the philanthropy section, could the following recent events be added?

 * "The KKFF manages the Pianos for Schools project, which raises money so that Czech schools can purchase new pianos. This project included the purchase of 11 Petrof pianos after a 200,000 euro order from a Chinese customer was cancelled in protest against Senate President Miloš Vystrčil’s visit to Taiwan."

 * https://praguebusinessjournal.com/komarek-buys-pianos-rejected-by-china-over-taiwan-row/
 * https://www.thetimes.co.uk/article/beijing-seeks-vengeance-after-czechs-back-taiwan-6bnd67gwd
 * https://www.taipeitimes.com/News/front/archives/2020/09/10/2003743138

 * "In June 2021, Karel Komarek offered financial aid to those affected by a tornado in South Moravia. The natural disaster caused huge long-lasting material damage in the area. Mr Komárek pledged a donation of 7 million USD to support the affected region."

 * https://www.seznamzpravy.cz/clanek/tornado-poskodilo-i-firmy-miliardare-komarka-na-hodoninsko-posle-penize-168305

Further request
In addition to the above, SAZKA Group rebranded to Allwyn last month. Can the article please be updated to reflect this by adding "Allwyn (previously known as SAZKA Group)" as appropriate throughout the article? I believe this is non-controversial so if there is no repsonse I will implement this myself. Panepani (talk) 16:55, 5 January 2022 (UTC)

 * https://www.sbcnews.co.uk/retail/2021/12/08/sazka-entertainment-renamed-allwyn-in-global-rebranding/
 * https://www.bbc.com/news/uk-politics-58971257

Update: this is now live on the page, removing 2nd request tag Panepani (talk)

Question
Hellow Edwardx,

I've been editing this article on Mr. Komarek's behalf. However, I do not fully understand why my edit was reverted. The previous version dealt mainly with his KKCG company, but my proposal focuses on Mr. Komarek as a person and his life, which I believe is an improvement of the article.

For example, his professional career has been significantly expanded in the "Early life" section, and more detailed information has been added. In the KKCG section, only essential information is included because there is a separate Wikipedia article about this company. In the Philantropy section I added some projects of his work within the foundations, as this is also additional information about him as a person.

Kind regards, --JamesCee1995 (talk) 12:42, 19 June 2023 (UTC)

 * Hello Edwardx, I haven't heard back from you in more than three weeks. Would you object if I reinstated my revision? And if so, what do you think needs improving to make my revision compliant with Wikipedia's guidelines and policies? Best regards, JamesCee1995 (talk) 07:05, 7 July 2023 (UTC)

title: Talk:Karel Komárek


text:
Geometry/Similar Triangles

For two triangles to be similar, all 3 corresponding angles must be congruent, and all three sides must be proportionally equal. Two triangles are similar if...
 * Two angles of each triangle are congruent.
 * The acute angle of a right triangle is congruent to the acute angle of another right triangle.
 * The two triangles are congruent. Note here that congruency implies similarity.

title: Geometry/Similar Triangles


text:
User:Carlschroedl

Hello!
I'm Carl Schroedl.

I am a software developer and an open-source civic hacker.

For more on me, visit my blog.

For more on my open-source civic hacking, see the Pivot Libre project.

title: User:Carlschroedl


text:
OIM

OIM may refer to:

 * Office for the Internal Market, UK internal trade body
 * OIM (offshore installation manager), an oil-rig worker
 * OIM, IATA code for Oshima Airport, in the island of Izu Ōshima, Tokyo, Japan
 * Tõnu Õim, correspondence chess player
 * Online Identity Management
 * Oracle Identity Management, an identity-management infrastructure for products from Oracle Corporation
 * Organisation internationale pour les migrations, the French name of International Organization for Migration
 * Orientation imaging microscopy

title: OIM


text:
Anne Frank and Me

Anne Frank and Me is a 2001 novel by husband and wife writing team Cherie Bennett and Jeff Gottesfeld. Inspired by the life of Anne Frank, it follows a teenage girl named Nicole Burns who travels back in time to 1942 and inhabits the body of a Jewish Holocaust victim. The novel was adapted from a play written and directed by Bennett in 1996.

Plot summary
Nicole Burns, a fifteen-year-old American high school student living in the year 2001, becomes fascinated with a Holocaust survivor named Paulette Littzer-Gold, who speaks to her English class. She feels they have met before. During a trip to a local Holocaust museum, Nicole and her peers are assigned roles as Jewish teens living during the Holocaust. Nicole is given the name Nicole Bernhardt. After the activity begins, Nicole hears students shrieking and gunfire. She attempts to run along with the rest of her classmates, but is struck in the back while ascending a staircase and loses consciousness.

Nicole wakes to find herself in 1942 Paris in the body of Nicole Bernhardt. Over the course of several months, she begins to forget her life in 2001 and internalizes her new identity. Several of Nicole's friends are non-Jews who oppose Adolf Hitler's policies and protect the Bernhardt family. Following the Nazi invasion of France, Nicole is forced to hide in a rundown apartment in the streets of Paris. From her refuge, Nicole writes a string of anti-Nazi letters for the French resistance. In the letters, she calls herself "GirlX" after the website she runs back home.

The Bernhardt family is betrayed and Nicole is transported to Auschwitz. Aboard the train, she meets Anne Frank. Nicole tells Anne she has read her diary, but Anne says she left her diary in the attic where she had been hiding. At Auschwitz, a fellow Jew tries to save Nicole by sending her to the labor camp instead of being sent to be killed. Nicole and her sister Liz-Bette, who is very ill, are to be split up, Nicole to live and Liz-Bette to die. Nicole becomes hysterical and begs to be allowed to accompany her sister. The Germans, after mocking Nicole's devotion to Liz-Bette, allow her to go with the young girl. Nicole tearfully thanks them and then walks with Liz-Bette to the "showers," where they recite a Jewish prayer before dying.

Nicole wakes up, lying on a bench outside the museum. She finds out that other students had set off firecrackers which sent everyone running, when she bumped her head. Nicole wonders if she really did go back in time or whether it was all a dream. After a few days' stay in the hospital, Nicole finds out Mrs. Littzer-Gold died overnight. She decides to go to Mrs. Littzer-Gold's funeral. After the funeral, Nicole notices that a letter Mrs. Littzer-Gold owned was one of the GirlX letters that Nicole herself had written, back in Paris in 1942. Nicole realizes that not only did she really experience the Holocaust firsthand, but she gave Mrs. Littzer-Gold the courage to survive.

Reviews
Reviews were mixed to negative. Kliatt called Anne Frank and Me a "powerful and affecting story" which "makes the deprivations and degradation of the Nazi occupation come alive." Publishers Weekly was not impressed, calling the novel "long on gimmickry and short on history." The reviewer pointed to "flimsy" writing and felt the time travel was "inconsistent and incompletely developed." Similarly, Kirkus Reviews accused the story of "pounding the reader over the head with the message" using "trite and forced" dialogue. Both periodicals compared it unfavorably with The Devil's Arithmetic by Jane Yolen.

title: Anne Frank and Me


text:
Redwall Friend & Foe

Redwall Friend & Foe was published in 2000 as an accessory to the Redwall series by Brian Jacques.

Summary
This guide features art by Chris Baker and contains descriptions of Redwall heroes and villains. It also features a pull-out poster and a number of questions to test the reader's knowledge of the series.

Heroes
The heroes featured in the guide are:
 * Martin the Warrior
 * Luke the Warrior
 * Sunflash the Mace
 * Mariel Gullwhacker
 * Joseph the Bellmaker
 * Urthstripe the Strong
 * Matthias
 * Mattimeo
 * Grath Longfletch
 * Tamello De Fformelo Tussock (Tammo)
 * Dannflor Reguba

Villains
The villains featured in the guide are:
 * Badrang the Tyrant
 * Tsarmina Greeneyes
 * Vilu Daskar
 * Swartt Sixclaw
 * Gabool the Wild
 * Urgan Nagru
 * Ferahgo the Assassin
 * Cluny the Scourge
 * Slagar the Cruel
 * Ublaz Mad-Eyes
 * Damug Warfang
 * Mokkan

title: Redwall Friend & Foe


text:
Talk:Soundtracks (Can album)

Fair use rationale for Image:Can-Soundtracks (album cover).jpg
Image:Can-Soundtracks (album cover).jpg is being used on this article. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in Wikipedia articles constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

If there is other other fair use media, consider checking that you have specified the fair use rationale on the other images used on this page. Note that any fair use images uploaded after 4 May, 2006, and lacking such an explanation will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.BetacommandBot 18:40, 1 June 2007 (UTC)

title: Talk:Soundtracks (Can album)


text:
Portal talk:Learning Materialsnew

Note: This is a "backup" copy of Portal:Learning Materials where new versions of the portal can be planned.

Colors
The page Portal:Learning Materialsnew/box-header contains the settings for the portal colors:
 * border=#aaaaaa
 * titleforeground=white
 * titlebackground=#aaccff
 * background=#f9f9ff
 * foreground=black

See w:Wikipedia:Portal/Instructions for details about how this portal format was created and can be modified.

Move
Going to move to Portal:Learning Materials/Concept so the "backup" development version can be kept with the main portal page.

title: Portal talk:Learning Materialsnew


text:
паства

Noun

 * 1) congregation, flock, parish

Noun

 * 1) herd

Noun

 * 1)  congregation, flock, parish

title: паства


text:
Wikipedia:Reference desk/Archives/Entertainment/2019 September 16

= September 16 =

Unsung British children's movie
In the early-to-mid 1980's, when I was in middle school, my classmates and I were shown a children's movie called The Salvage Gang. I just recently rediscovered it on YouTube. The film, released in 1958, takes place in London. Why isn't there an article about it on Wikipedia?2604:2000:7104:2F00:85A7:C1A8:90C4:85A6 (talk) 09:42, 16 September 2019 (UTC)
 * It's mentioned in this article, but there isn't an article on the film itself. Everyone here is a volunteer, so feel free to start writing one.   Ghmyrtle (talk) 09:57, 16 September 2019 (UTC)
 * As you can see from the article linked above, the Children's Film Foundation made an awful lot of very low-budget films, often with otherwise unknown actors. So the problem with writing articles about them might be finding reliable sources. Alansplodge (talk) 10:36, 20 September 2019 (UTC)
 * The OP probably lacks (for now) the ability to create an article. —Tamfang (talk) 20:24, 21 September 2019 (UTC)

Vern Fleming DoB
The article Vern Fleming says he was born on born February 4, 1962; the infobox says February 4, 1961. Given that his brother Victor Fleming (basketball) was born on February 2, 1962, then 1961 seem the more likely date for Vern. Can anyone give a reliable citation for this, please? Andy Mabbett ( Pigsonthewing ); Talk to Andy; Andy's edits 10:28, 16 September 2019 (UTC)
 * Neither article has a reference for any birthdate. All of the dates should be removed unless and until a reliable source can be produced for anything.  -- Jayron 32 12:30, 16 September 2019 (UTC)
 * Could the Fleming brothers be twins with different birthdates due to complications of delivery? --Khajidha (talk) 16:47, 16 September 2019 (UTC)
 * They could also be adopted brothers from different birth parents with nearby birthdates. But none of that matters because without a source, it should be removed.  -- Jayron 32 17:30, 16 September 2019 (UTC)
 * There are lots of reliable sources for Vern and 1962, so I'm going to put that back. Clarityfiend (talk) 19:51, 16 September 2019 (UTC)


text:
Talk:Girlfriend in a Coma (novel)

Karen Ann Quinlan
Any chance the character is based off of Karen Ann Quinlan? --Twintone 18:36, 11 August 2006 (UTC)
 * Yes indeed, see Talk:Karen Ann Quinlan. -- FP (talk)(edits) 03:28, 28 October 2006 (UTC)

Smiths references
We might want to mention the continual Smiths references scattered through the book (e.g. when Karen is having a vision of the burial at Windsor and says "The Queen is dead.") Not a list, just a mention. Marnanel 14:00, 10 July 2007 (UTC)
 * Done.--Twintone 16:49, 12 July 2007 (UTC)

Fair use rationale for Image:Girlfriensincoma.jpg
Image:Girlfriensincoma.jpg is being used on this article. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in this Wikipedia article constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

BetacommandBot (talk) 06:45, 1 January 2008 (UTC)

External links modified
Hello fellow Wikipedians,

I have just added archive links to 1 one external link on Girlfriend in a Coma (novel). Please take a moment to review my edit. If necessary, add after the link to keep me from modifying it. Alternatively, you can add to keep me off the page altogether. I made the following changes:
 * Added archive https://web.archive.org/20130326220547/http://www.imdb.com:80/title/tt2698758/ to http://www.imdb.com/title/tt2698758/

Cheers. —cyberbot II  Talk to my owner :Online 04:50, 18 October 2015 (UTC)

title: Talk:Girlfriend in a Coma (novel)


text:
stich

Etymology 1
. Akin to.

Noun

 * 1)  A verse, of whatever measure or number of feet, especially a verse of Scripture.
 * 2) A part of a line of poetry, especially in the distichal poetry of the Hebrew Bible and in early Germanic heroic verse such as Beowulf, where the line is composed of two (occasionally three) such parts.
 * 3)  A row, line, or rank of trees.

title: stich


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Hammons Student Center

The John Q. Hammons Student Center is an 8,846-seat multi-purpose arena on the campus of Missouri State University in Springfield, Missouri. It was built in 1976 and is the home of the Missouri State Bears. The arena was replaced by the Great Southern Bank Arena in 2008.

title: Hammons Student Center


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Simone Ciulli

Simone Ciulli (born 9 May 1986) is an Italian swimmer who competes in Paralympic S9 events. and who has been a member of the Italian National Swim Team since 2013. In August 2021, he won a silver medal at the XVI Paralympic Games in Tokyo 2020. In September 2019, he won a bronze medal at the Paralympic World Swimming Championship in London. He has also won a silver at the Mediterranean Games of Mersin in 2013 and two bronze medals at the European Championships in Funchal in 2016.

Career
Since birth, Ciulli has had cerebral palsy. This has caused him a half-paresis on the right side of his body and from perineuroma which is located on his right sciatic nerve.

In 2012, he was introduced to the Paralympic organization and entered the world of Paralympic swimming by participating at the Italian Paralympic Swimming Championships in Pietralata, with Po.Ha.Fi. team. There, he was first classified as an S10, SM10 and SB9. At that meet, he broke two Italian records in the 50 and 100 freestyle.

In 2013, he was invited for the first time to join the Italian national Paralympic swimming team and competed for the red, white and green at his first international meet in Berlin.

A few months later, he was invited, together with the paralympic champion Federico Morlacchi, to compete at the XVII Mediterranean Games, in Mersin Turkey, where he won the silver medal in the 100 freestyle with a time of 57 "14.

That same year he was selected by the National Team to take part in the FINA World Cup for non-disabled swimmers in Eindhoven. There he won three gold medals in the 50 and 100 freestyle and the 100 IM. At that same event, he competed together with non-disabled athletes from the Italian National Swimming Team including, Gregorio Paltrinieri, Fabio Scozzoli, Ilaria Bianchi and Federico Turrini.

In August 2014 at the European Paralympic Swimming Championships in Eindhoven, he broke the Italian S10 long course national record with a time of 25"18, while placing fourth in the event. In Eindhoven, he also set the Italian S10 national backstroke record in the 4x100 mixed relay with a time of 1'04"69.

In 2015, he once again was invited to join the National swim team to compete in an international meet in Berlin where he established two new Italian national S10 records in the 100 freestyle with a time of 56"35 and in the 100 butterfly, with a time of 1'00"88. These times were just a few tenths of a second away for the qualifying time for 2016 Paralympics in Rio de Janeiro.

In 2016 at the European Championships in Funchal he won two bronze medals. One in the 50 freestyle with a time of 25"40 and the other in the 4x100 mixed relay.

In 2017 at the Italian Winter Paralympic Swimming Championships held in Portici, he became the first Italian Paralympic athlete to swim the 50 freestyle under 25 seconds (24"89) and the 100 freestyle under 55 seconds (54"77), also establishing the new national short course records.

At the 2019 World Series in Lignano, he was reclassified as an S9, SM9 SB8 swimmer because the tumor located on his right sciatic nerve grew and progressively hampered the functionality of his lower right leg.

In September 2019, he represented Italy at the World Paralympic Swimming Championships in London, where he won a bronze medal in the 50 metres freestyle with a time of 26"04 and he became World Champion with the National Team of Italy.

In August 2021, he was called to XVI Paralympic Games in Tokyo 2020, winning a silver medal in the 4x100 freestyle relay with the time of 3 '.45'' .89 that was a new European Record.

Extra-competitive activities
In 2011 he graduated with a master's degree in Law and subsequently, in 2015 passed the Bar exam to earn his license to practice. He has elected to not pursue his law career until he retires from competitive swimming.

Concurrently in 2014, he obtain a number of state certifications to expand his expertise in swimming and water safety. His certifications include, life guard, second level triathlon trainer, water polo instructor, various fitness instructor, swim school coordinator, as well as sports and facilities management.

Since 2010, he has also been the head coach Klab Nuoto Master Swim Team in Florence, Italy. He has built the program from scratch currently counts over 100 athletes:, Klab Nuoto Masters has won six consecutive Italian Master Swimming Championships U.I.S.P.

Additionally, since 2016 he has been the personal trainer of Noura Mana, who is a member of the Moroccan National Swimming Team and who participated in the Olympic Games of Rio de Janeiro in 2016. Under Ciulli's guidance, she has broken two Moroccan national records, in the 50 freestyle (25 and 50 metres). She has won a bronze medal at the 2019 Pan-African Games in Casablanca and has also participated at the World Short Course Championships in 2018 in Hangzhou.

title: Simone Ciulli


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Talk:Carl Westcott

Chart Westcott
1) This article is not about Chart Westcott. 2) Your citation does not support a claim that Chart Westcott is an alcoholic, and that claim is a violation of Wikipedia's Biography of Living Persons policies. 3) Your citation indicates that Chart Westcott has been charged with a misdemeanor. That does not make him a "fugitive." This is another WP:BLP violation.  If you continue to insert this information, which must be removed as a WP:BLP violation, you will be blocked.   Dwpaul   Talk   20:33, 17 May 2014 (UTC)
 * The State Bar of Texas seems to differ with you. TXRD18 (talk) 20:42, 17 May 2014 (UTC)
 * Nothing on the Texas State Bar page mentions alcohol or alcoholism. Therefore it is not a valid source to support your claim.  Dwpaul  Talk   20:46, 17 May 2014 (UTC)

title: Talk:Carl Westcott


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European History/World War I

As a result of German unity and increasing German nationalism, as well as various other causes, Germany began on what Kaiser Wilhelm II called a "new course" to earn its "place in the sun." After 1871, Germany's trade and industry increased vigorously, challenging and, in some areas, even exceeding that of Great Britain, until then the premier industrial nation of Europe. A many-sided rivalry developed between Germany and Britain, intensifying when the sometimes-bellicose Wilhelm II assumed power and began building a strong, ocean-going navy.

Seeking to balance the rise of German power, Britain and France began to draw closer together diplomatically as the 20th century began. Germany, meanwhile, had allowed an implicit alliance with Tsarist Russia to lapse, and faced ongoing French resentment over the provinces of Alsace and Lorraine which Germany had annexed in 1871. The perceived danger of "encirclement" by hostile nations began to loom in the minds of German leaders. These factors together formed some of the tinder which would ignite the outbreak of war in 1914.

It is interesting to note, however, that all the ruling families of Europe were related to each other in some form or fashion. This led to many Europeans feeling that it was a family affair that they had been dragged into and forced to endure.

The Road to War
World War One is one of the most hotly contested issues in history; the complexity and number of theorized causes can be a major cause for confusion. One of the main reasons for this complexity is the long period over which this war’s tension built, beginning with the unification of Germany by Bismarck, and escalating from there on. There is no doubt that Germany's misguided foreign policy contributed to the outbreak of war, however the extent to which it contributed is the contended issue.

Some historians suggest that Germany willed the war and engineered its outbreak, and others even suggest that Germany felt compelled to go to war at that time. However, some suggest that the war was brought about by poor leadership at the time, others argue that the war was brought about by accident - that Europe stumbled into war due to tension between alliance systems. Finally, some historians argue that World War I was the culmination of historical developments in Europe. This argument states that war was inevitable between Austria and Serbia, that imperial expansion by Russia eastward was also likely to provoke war, and that the French were still furious over their loss of Alsace-Lorraine in the Franco-Prussian war.

There was certainly a general rise in nationalism in Europe, which played a major role in the start of the conflict. The war became inevitable when the so-called "blank check" was created when Austro-Hungarian Emperor Franz Joseph sent a letter to German Kaiser Wilhelm II, asking for German support against Serbia. Theobald von Bethmann-Hollweg, Kaiser Wilhelm II's Imperial Chancellor, telegrammed back that Austria-Hungary could rely on Germany to support whatever action was necessary to deal with Serbia.

The Alliance System
One factor which helped to escalate the conflict was the alliance system of the late nineteenth century. Although there had always been alliances between different European nations, the diplomatic trend during the nineteenth century was to have secret alliances, committing states to defensive military action. These were encouraged by Bismarck who, in the process of unifying Germany, sought to pacify those surrounding states which might proved hostile. Although Germany had originally allied itself with the empires of Austria and Russia at this time, by the beginning of the twentieth century alliances had shifted. Germany found itself allied with Austria-Hungary and Italy, the so-called "Central Powers"; together the countries formed what was known as the "Triple Alliance". Meanwhile, France, desperate for allies after the Franco-Prussian war, had cultivated a friendship with Russia. Great Britain, too, felt isolated in the increasingly factionalised European environment, and sought out an alliance with another of Europe's great powers. This led to the Entente Cordial with France, which was to develop into the "Triple Entente" between Britain, France and Russia.

By 1914 most of the smaller European states had been drawn into this web of alliances. Serbia had allied itself with Russia whilst its enemy, Bulgaria, chose the patronage of Germany. A number of small states maintained their neutrality in this complicated network of alliances. Belgium, for instance, was a neutral state, its independence guaranteed by Britain, France, and Germany.

The Schlieffen Plan

The Schlieffen Plan was designed by Field Marshall Count Alfred von Schlieffen, who became Chief of the Great General Staff in 1891 and submitted his plan in 1905. Out of fear of a two front war, which Germany was nearly certain it could not win, it devised the plan to eliminate one of the fronts of the war before the other side could prepare. The plan called for a rapid German mobilization, sweeping through Holland, Luxembourg and Belgium into France. Schlieffen called for overwhelming numbers on the far right flank, the northernmost spearhead of the force with only minimum troops making up the arm and axis of the formation as well as a minimum force stationed on the Russian eastern front. Swift elimination of the French threat would in theory allow Germany to better defend against a Russian, or a British force. However, the British involvement was not looked for under the Schlieffen plan, not at the commencement of action at least. In 1905 Count Schlieffen expected his overpowering right wing to move basically along the coast through Holland. He expected the Dutch to acquiesce and grant the army the right to cross their borders. Schlieffen knew that navigating around the Belgian fortress at Liege in this way would speed the advance while still defeating the fortress simply by encirclement. Schlieffen retired from his post in 1906 and was replaced by Helmuth von Moltke. In 1907-08 Moltke adjusted the plan, reducing the proportional distribution of the forces, lessening the crucial right wing in favor of a slightly more defensive strategy. Also, judging Holland as unlikely to grant permission to cross its borders the plan now called for a direct move through Belgium but expected the French force to officially invade neutral Belgium first in an attempt to take the advantageous position at Meuse. Moltke's variation called for an artillery assault on Liege, but with the rail lines and the unprecedented firepower the German army brought what he did not expect any significant defense of the fortress.

August 1914: War Erupts
On June 28, 1914, Archduke Franz Ferdinand, heir to the Austria-Hungary throne, was assassinated in Sarajevo. As a result, Austria declared war on Serbia. Germany declared war on both Russia and France. On August 4, Germany invaded neutral Belgium before the French. This precipitated in Great Britain's declaration of war on Germany.

Enacting the Moltke variation of the Schlieffen Plan, German forces entered Belgium, attacking the fortress of Liege. Although they could not stop the large invading force, Belgian troops fought bravely, and the siege lasted 10 days, arguably upsetting the German timetable and allowing for mobilization of the French and the British Expeditionary forces. During the second half of August, however, a hasty French counteroffensive in Lorraine collapsed, with heavy casualties in the face of German machine-gun fire. French armies fell back in disarray as the Germans crossed from Belgium into France on a wide front.

Keeping its alliance with France, Russia's armies invaded Germany's easternmost province, East Prussia, in August. The German high command dispatched General Paul von Hindenburg to defend the province. Hindenburg took command and defeated the Russians at the Battle of Tannenberg, ending the hope of a Russian advance to Berlin.

The end of August was marked by near-panic in northern France as the German offensive rolled south toward Paris, seemingly unstoppable. On the German side, however, a gap developed between the westernmost army corps, and the rapid advance was exhausting the troops. The French rushed reinforcements from Paris—some in taxicabs—to the front, and by the first week of September, amid heavy fighting, the Germans had been halted along the River Marne. This marked the beginning of the static trench lines which would define the front in Western Europe for four years.

1915-1916
On February 4, 1915, Germany declared a submarine blockade of Great Britain, stating that any ship approaching England was a legitimate target. On May 7, 1915, Germany sank the passenger ship Lusitania, resulting in a massive uproar in the United States, as over 100 U.S. citizens perished. On August 30, Germany responded by ceasing to sink ships without warning.

The front in France became the focus of mass attacks that cost huge numbers of lives, but gained very little. Britain became fully engaged in France, raising a large conscript army for the first time in its history. 1915 saw the first attacks with chlorine gas by the Germans, and soon the Allies responded in kind. During much of the year 1916, the longest battle of the war, the Battle of Verdun, a German offensive against France and Britain, was fought to a draw and resulted in an estimated one million casualties. On July 1 through November 18, the Battle of Somme, a British and French offensive against the Germans, again resulted in approximately one million casualties but no breakthrough for either side.

1917-1918: Final Phases
On February 1, 1917, Germany again declares unrestricted submarine warfare. The Germans believed that it was possible to defeat the British in six months through this, and assumed it would take at least one year for America to mobilize as a result of the actions. Thus, they banked on the hope that they could defeat Britain before America would enter the war.

A mood of cultural despair had settled over much of Europe by this time, as an entire generation of young men was fed into the maw of combat. French armies came close to mutiny in 1917 when ordered into an attack they knew would be hopeless. Germany, blockaded from overseas trade, saw hunger and deprivation among the population, with labor strikes and political discontent growing. Russia underwent collapse, its armies defeated and the Tsar ousted in favor of a liberal-socialist regime.

On April 6, 1917, the United States entered the war by declaring war on Germany. This was in part due to the sinking of the Lusitania and the Zimmerman Telegram, which was a ploy to convince Mexico to attack the United States in exchange for the return of Texas, New Mexico, and Arizona should the United States enter the war. From July 31 through November 10, 1917, the Third Battle of Ypres, also known as Passchendaele, resulted in minor gains for the British, but there was still no breakthrough of the well-developed German defenses. During this time, on November 7, Bolsheviks, led by Lenin, overthrew the post-Tsar Russian government.

As a result, in March 1918, the new Russian government, represented by Leon Trotsky, signed an armistice treaty with Germany, removing the eastern front of the war for Germany. On March 21, Germany thus launched what is known as the Ludendorff offensive in the hope of winning the war before American troops arrived. The final German effort, however, fared no better in the end than the previous ones; the Germans pushed closer to Paris than ever before, but by the end of summer they had exhausted themselves against the Allied defenses, now including fresh American armies.

On September 29, 1918, allied troops broke through the German fortifications at the Hindenberg line, and the end of the war came into view. On November 9, 1918, Kaiser Wilhelm II abdicated, and on November 10 the German Weimar Republic was founded. On November 11, 1918, at eleven o'clock on the eleventh day of the eleventh month of 1918, the war ended as Germany and the Allies signed an armistice agreement.

The War in Italy
Italy had been allied with the German and Austro-Hungarian Empires since 1882 as part of the Triple Alliance. However, the nation had its own designs on Austrian territory in Province of Trento, Istria and Dalmatia. Rome had a secret 1902 pact with France, effectively nullifying its alliance. At the start of hostilities, Italy refused to commit troops, arguing that the Triple Alliance was defensive in nature, and that Austria-Hungary was an aggressor. The Austro-Hungarian government began negotiations to secure Italian neutrality, offering the French colony of Tunisia in return. The Allies made a counter-offer in which Italy would receive the Alpine province of South Tyrol and territory on the Dalmatian coast after the defeat of Austria-Hungary. This was formalised by the Treaty of London. Further encouraged by the Allied invasion of Turkey in April 1915, Italy joined the Triple Entente and declared war on Austria-Hungary on May 23. Fifteen months later Italy declared war on Germany.

Militarily, the Italians had numerical superiority. This advantage, however, was lost, not only because of the difficult terrain in which fighting took place, but also because of the strategies and tactics employed. Field Marshal Luigi Cadorna, a staunch proponent of the frontal assault, had dreams of breaking into the Slovenian plateau, taking Ljubljana and threatening Vienna. It was a Napoleonic plan, which had no realistic chance of success in an age of barbed wire, machine guns, and indirect artillery fire, combined with hilly and mountainous terrain.

On the Trentino front, the Austro-Hungarians took advantage of the mountainous terrain, which favoured the defender. After an initial strategic retreat, the front remained largely unchanged, while Austrian Kaiserschützen and Standschützen engaged Italian Alpini in bitter hand-to-hand combat throughout the summer. The Austro-Hungarians counter-attacked in the Asiago towards Verona and Padua, in the spring of 1916, but made little progress.

Beginning in 1915, the Italians under Cadorna mounted eleven offensives on the Isonzo front along the Isonzo River, north-east of Trieste. All eleven offensives were repelled by the Austro-Hungarians, who held the higher ground. In the summer of 1916, the Italians captured the town of Gorizia. After this minor victory, the front remained static for over a year, despite several Italian offensives. In the autumn of 1917, thanks to the improving situation on the Eastern front, the Austrians received large numbers of reinforcements, including German Stormtroopers and the elite Alpenkorps. The Central Powers launched a crushing offensive on 26 October 1917, spearheaded by the Germans. They achieved a victory at Caporetto. The Italian army was routed and retreated more than 100 km (60 miles) to reorganise, stabilizing the front at the Piave River. Since in the Battle of Caporetto Italian Army had heavy losses, the Italian Government called to arms the so called '99 Boys (Ragazzi del '99), that is, all males who were 18 years old. In 1918, the Austro-Hungarians failed to break through, in a series of battles on the Asiago Plateau, finally being decisively defeated in the Battle of Vittorio Veneto in October of that year. Austria-Hungary surrendered in early November 1918.

The War at Sea
At the start of the war, the German Empire had cruisers scattered across the globe, some of which were subsequently used to attack Allied merchant shipping. The British Royal Navy systematically hunted them down, though not without some embarrassment from its inability to protect Allied shipping. For example, the German detached light cruiser SMS Emden, part of the East-Asia squadron stationed at Tsingtao, seized or destroyed 15 merchantmen, as well as sinking a Russian cruiser and a French destroyer. However, the bulk of the German East Asia Squadron — consisting of the armoured cruisers SMS Scharnhorst and SMS Gneisenau, light cruisers SMS Nürnberg and SMS Leipzig and two transport ships — did not have orders to raid shipping and was instead underway to Germany when it encountered elements of the British fleet. The German flotilla, along with SMS Dresden, sank two armoured cruisers at the Battle of Coronel, but was almost completely destroyed at the Battle of the Falkland Islands in December 1914, with only Dresden and a few auxiliaries escaping, but at the Battle of Más a Tierra these too were destroyed or interned.

Soon after the outbreak of hostilities, Britain initiated a naval blockade of Germany. The strategy proved effective, cutting off vital military and civilian supplies, although this blockade violated generally accepted international law codified by several international agreements of the past two centuries. Britain mined international waters to prevent any ships from entering entire sections of ocean, causing danger to even neutral ships. Since there was limited response to this tactic, Germany expected a similar response to its unrestricted submarine warfare.

The 1916 Battle of Jutland (German: Skagerrakschlacht, or "Battle of the Skagerrak") developed into the largest naval battle of the war, the only full-scale clash of battleships during the war. It took place on 31 May–1 June 1916, in the North Sea off Jutland. The Kaiserliche Marine's High Seas Fleet, commanded by Vice Admiral Reinhard Scheer, squared off against the Royal Navy's Grand Fleet, led by Admiral Sir John Jellicoe. The engagement was a standoff, as the Germans, outmaneuvered by the larger British fleet, managed to escape and inflicted more damage to the British fleet than they received. Strategically, however, the British asserted their control of the sea, and the bulk of the German surface fleet remained confined to port for the duration of the war.

German U-boats attempted to cut the supply lines between North America and Britain. The nature of submarine warfare meant that attacks often came without warning, giving the crews of the merchant ships little hope of survival. The United States launched a protest, and Germany modified its rules of engagement. After the notorious sinking of the passenger ship RMS Lusitania in 1915, Germany promised not to target passenger liners, while Britain armed its merchant ships, placing them beyond the protection of the "cruiser rules" which demanded warning and placing crews in "a place of safety" (a standard which lifeboats did not meet). Finally, in early 1917 Germany adopted a policy of unrestricted submarine warfare, realizing the Americans would eventually enter the war. Germany sought to strangle Allied sea lanes before the U.S. could transport a large army overseas.

The U-boat threat lessened in 1917, when merchant ships entered convoys escorted by destroyers. This tactic made it difficult for U-boats to find targets, which significantly lessened losses; after the introduction of hydrophone and depth charges, accompanying destroyers might actually attack a submerged submarine with some hope of success. The convoy system slowed the flow of supplies, since ships had to wait as convoys were assembled. The solution to the delays was a massive program to build new freighters. Troop ships were too fast for the submarines and did not travel the North Atlantic in convoys. The U-boats had sunk almost 5,000 Allied ships, at a cost of 178 submarines.

World War I also saw the first use of aircraft carriers in combat, with HMS Furious launching Sopwith Camels in a successful raid against the Zeppelin hangars at Tondern in July 1918, as well as blimps for antisubmarine patrol.

New Military Techniques and Technologies

World War I introduced the first time that total war was employed - that is, the full mobilization of society occurred in participant nations. In addition, it marked the end of war as a "glamorous occupation," showing how brutal and horrifying war could be when fought by industrial nations with mass production of weapons, and mass armies drawn from whole populations.

World War I introduced a number of new technologies and techniques. The outbreak of war took the world from the age of coal to an age where energy was largely derived from petroleum, a much higher-grade fuel source used in many new fighting machines and transport systems on land and sea. The deadliest product of this new industry was chemical warfare, with countless fighting men suffering and dying in gas attacks. Submarines also were used with effect, leading to the advent of depth charges and sonar. Rudimentary tanks and mechanized warfare also entered the battlefield near the end of the war. Finally, the machine gun took its toll for the first time in World War I. All this was aimed at breakthrough in trench warfare, in which both sides would dig deep trenches, and attempt to attack the other side, most often with little or no success.

The Russian Revolution
The Russian Revolution marked the first outbreak of communism in Europe. Contrary to popular belief, however, there were in fact two specific and unique revolutions that took place during 1917 - a true Marxist revolution as well as a revolution led by Lenin that was not a true Marxist revolution.

March Revolution of 1917
The peasants were unhappy with the czar as a result of losses from World War I, the lack of real representation and the czar's dismissal of the Duma, the influence of Rasputin upon Alexandra, hunger, food shortages, and industrial working conditions.

As a result, on March 8, 1917, food riots broke out in St. Petersburg; however, the soldiers refused to fire upon the rioters. At this time, two forces were in competition for control of the revolution. Members of the Duma executive committee called for a moderate constitutional government, while Soviets, members of worker councils, pushed for revolution and industrial reform.

On March 15, 1917, the Czar attempted to return to Russia by train, but was stopped by the troops and was forced to abdicate.

From March through November, a provisional government was led by Alexander Kerensky, a socialist, and Prince Lvov. However, this government was destined to fail because it took no action in land distribution, continued to fight in World War I, and failed to fix food shortages. General Kornilov attempted a coup, but Kerensky used the Soviets and Bolsheviks to put down the coup. However, this action showed the weakness of Kerensky.

The March Revolution marked the first time that the class struggle predicted by Karl Marx took place. Thus, the March Revolution was a true Marxist revolution based upon the theories of Marx in The Communist Manifesto.

November Revolution of 1917

Vladimir Lenin realized that the time had come to seize the revolution. He authored the "April Theses," in which he promised peace with the Central Powers, redistribution of the land, transfer of the factories to the owners, and the recognition of the Soviets as the supreme power in Russia. In this sense, the November Revolution was led by Lenin rather than being an overall coup by the workers, and thus the November revolution cannot be dubbed a true Marxist revolution.

The Revolution may never have happened had not the Prime Minister of the Time, Aleksandr Kerensky, destroyed the power of authority within the Army and Navy by allowing the Bolshevik and Menshevik committees greater powers. Kerensky effectively disarmed the Man who could have prevented the Revolution ever happening. The man in question, was the Commander-in-Chief, General Lavr Kornilov who attempted to bring to heel the populist Government of Kerensky and instill some authority back in to the state and the Army. Kerensky seized the opportunity to relieve Kornilov of his office and effectively gave the Bolsheviks, namely the Red Guard within the ranks of the Petrograd sailors, a Carte Blanc to take up arms in the so called defense of the Provisional Government. The Army lost its Commander and the streets were handed over to the Bolsheviks.

In March 1918 Lenin established the "dictatorship of the proletariat," adopted the name "Communist Party," and signed the treaty of Brest-Litovsk, withdrawing Russia from World War I.

Civil war raged in Russia from 1918 until 1922, pitting the Reds (Bolsheviks led by Trotsky) against the Whites, which consisted of czarists, liberals, the bourgeois, Mensheviks, the U.S., Britain, and France.

The victorious Bolsheviks acted to eliminate their opposition using secret police groups such as the Cheka, the NKVD, and the KGB. Lenin attempted to maintain Marxism, hoping to reach Marx's state of a propertyless, classless utopia. However, the pursuit of communism generally failed and the economy declined. Accordingly, Lenin enacted the New Economic Policy in March 1921, which compromised many aspects of communism for capitalism's profits.

Modern Art
The 1900s led to the creation of the new, modern art movement. Fauvism is a type of modern art that emphasized wild, extreme colors, abstraction, simplified lines, freshness, and spontaneity. Cubism is another form of modern art, which utilizes a geometrical depiction of subjects with planes and angles. The modern art movement arose because, with the advent of photography, art subjects no longer needed to be a realistic portrayal.

Perhaps the most famous modern artist is Pablo Picasso, a Spanish painter and sculptor.

title: European History/World War I