TITLE: Topology $\text{i})$ What is a topology? $\text{ii})$ What does a topology induced by a metric mean?
QUESTION [6 upvotes]: I am now trying to understand what a topology and a topological space is.
Yes, I know the "formal" or "mathematical definition" of it, it is in my notes so it's easy for me to reiterate that. Please bear with me as I am trying my best to express my confusion, it's a little hard to even do that in words.
Here's the definition I am sticking to

$X$ is a set. A topology on X is a set of subsets $\tau$ of $X$ with the following properties

Whenever $(U_{i})_{i \in I}$ is a family(finite or not) of subsets of $X$ such that $U_i \in \tau$, $\forall i \in I$ then $\cup_{i \in I}U_i \in \tau$

Whenever $U_1$, $U_2 \in \tau$ then $U_1 \cap U_2 \in \tau$.

$\phi \in \tau$ and $X \in \tau$



$i$) So my problem is,for each 1,2,3 conditions, I understand what they mean. So each subset in $\tau$ satisfies 1,2,3, which as a whole, is some subset of $X$ and we'll just name it "a topology" on $X$...yes?
My issue is, I don't see what this "collectively" gives. So if you are to explain it to someone who does not really do math, and explain it casually, what would you say? what does this "set of subsets" give? Is it nothing more than "just the set of subsets that satisfy 1,2,3, end of story"?
That is probably one of the reason why I cannot find a topology for any specific given set. I just don't know how.
$ii$) And also, the notion of induced topology by a metric is unclear to me.
The "discrete topology" as I hear very often, is apparently one of the most common topology which is induced by the discrete metric.
In my notes, it says

The discrete topology of $X$ is the collection of all subsets of $X$, which is the largest possible topology on $X$.

So, if I take a set of positive integers $X=\mathbb{Z}^+$ then the discrete topology is $\{\mathbb{Z}^+, \phi,\{1\},\{2\},\{3\},...,\{1,2\},\{1,3\},...,\{2,3\},\{2,4\},...,\{1,2,3\},\{1,2,4\},...\}$?
But even if it is, how is this "induced by the discrete metric? How is it relevant? I could have just blindly followed the definition of a topology to get all these subsets...right? without using or referring to any metric.
$iii$) And what is a topological space? And yes, I "know" it's $(X, \tau)$ where $\tau$ is a topology on $X$ but again, I can't help but confuse when I think of "metric spaces"
Metric space so far makes a lot more sense to me. I see $d(x,y)$, a metric as a function, so when we say a metric space $(X,d)$ just as how we phrase $(X,\tau)$,I understand it as a set $X$ with some function$d(x,y)$ which can be "applied" to the elements of $X$ to "give a value", which is the "distance."
So it's a set $X$ where I have given a "method" to tell how far any 2 elements are in it.
Now, $(X, \tau)$ doesn't sink in to me because $\tau$, unlike $d(x,y)$ doesn't do anything to $X$ (does it?). Meaning, it's just a set in a sense(with some special features) but does not allow me to "pick some value in $X$" to give me "some value"(whatever it is). So it comes to, as stupid as it may sound, "what is the point of having $\tau$"?
Ultimately...just what is a "topological space"? A pair of sets $X$ and $\tau$?
If it is analogous to the metric space, $d(x,y)$ "defined" on $X$, what would it mean that $\tau$ "defined" on $X$?
I hope expert topologists would understand what I am confused with an where I have the wrong way of thinking. I really need someone to enlighten me here, it's just getting so abstract to me..
I really appreciate your reading until here, it was indeed a long question, I am sorry. Thank you in advance for you help

REPLY [2 votes]: Let's pretend there's a such thing as "infinitely close." (There isn't, generally, but let's pretend.) We care about this, because ideas like "continuity" depend on an intuitive notion of "infinitely close," even though it doesn't actually exist (usually). So, how would we describe this notion?
Well, with metric spaces, if we want to say that $x$ and $y$ are infinitely close, we can try saying something like "$d(x,y)<1$, and $d(x,y)<0.1$, and $d(x,y)<0.01$, and…" and so on. That is, they approach it with the idea of distance.
Topological spaces approach this from a more set-theoretic perspective. If we want to say $x$ is infinitely close to $0$, we could say "$x\in(-1,1)$, and $x\in(-0.1,0.1)$, and $x\in(-0.01,0.01)$, and…"
This line of thinking leads you to the idea of a "neighborhood." (A set $N$ is a neighborhood of $x$ if there's an open set $O$ with $x\in O\subseteq N$.) In this line of thinking, a neighborhood of $x$ is expected to contain everything infinitely close to $x$. Most people introduce topology in terms of open sets, mostly out of convenience. (An open set is a set that's a neighborhood of each of its points.) However, I've also seen it introduced in terms of neighborhoods. It doesn't really matter, since you can define them in terms of each other.
So, the topology of a topological space is the set of open sets in it. The axioms of the topology are the properties that we'd expect open sets to have. (There are some properties that almost made the list, but people opted for generality and decided that they aren't necessary for topologies. I'm thinking of the so-called separation axioms, which you may learn about later.)