TITLE: Interesting Proofs about Metric Spaces?
QUESTION [12 upvotes]: I'm currently working through the book Introduction to Topology by Bert Mendelson, and I've finished all of the exercises provided at the end of the section that I have just completed, but I would like some more to try. I've just finished learning about metric spaces, continuity, and open balls about points in metric spaces. Just for a bit of context, some of the proofs that I have done include:


If $(X,d)$ is a metric space and $a\in X$, for each $\delta \gt 0$, the open ball $B(a; \delta)$ is a neighborhood of each of its points.
Let $f:(X,d)\to (Y,d')$, $a\in X$, and let $\beta_{f(a)}$ be a basis for the neighborhood system at $f(a)$. Prove that $f$ is continuous at $a$ iff $f^{-1}(N)$ is a neighborhood of $a$ for each $N \in \beta_{f(a)}$.
If $a$ and $b$ are distinct points of a metric space $X$, prove that there exist neighborhoods $N_a$ and $N_b$ of $a$ and $b$ respectively such that $N_a \cap N_b=\varnothing$.
If $(X,d)$ is a metric space containing $a$ and $b$, and $\delta+\eta \lt d(a,b)$, then $B(a;\delta)\cap B(b;\eta)=\varnothing$.


Can anybody give me any other (perhaps slightly more challenging) proofs to do about these topics? I would like to practice some more with them, but I'm not very good about forming true conjectures to prove.
Thanks!

REPLY [3 votes]: (1.1). Show that if $F$ is a family of subsets of a metric space such that $\cup G$ is closed whenever $G$ is a countable subset of $F$ , then $\cup F$ is closed.
(2). Different metrics that generate the same topology are called equivalent metrics:
(2.1). Show that if $d,e$ are equivalent metrics on  $X$    iff for every $r>0$ and every $x\in X$ there exist $r'>0$ and $r''>0$ such that $B_d(x,r')\subset B_e(x,r)$ and $B_e(x,r'')\subset B_d(x,r).$
(2.2). Let $d,e$ be metrics on $X$ such that there exist positive $k,k'$ such that $d(u,v)\leq k\cdot e(u,v)$ and $e(u,v)\leq k'\cdot d(u,v)$ for all $u,v \in X.$ Show that  $d,e$ are equivalent. (If such $k,k'$ exist then $d,e$ are called uniformly equivalent). And give an example of two equivalent metrics that are not uniformly equivalent.
(2.2).  For a  metric $d,$ show that $e_1=d/(1+d)$ and $e_2=\min (1,d)$  are metrics and are  equivalent to $d.$
(2.3). Let $(X,d)$ and $(Y,e)$ be metric spaces and let  $f:X\to Y$ be continuous. Show that $$f(u,v)=d(u,v)+e(f(u),f(v)) \quad \text {for } u,v\in X$$ is a metric on $X$ equivalent to $d.$ (In particular, with $Y=\mathbb R$ and $e(y,y')=|y-y'|,$ this is useful in constructions for other problems and examples.)
(3). Sequences and closures:
(3.1). In a metric space $(X,d)$ with $x\in X,$ show that a sequence $(x_n)_{n\in \mathbb N}$ of members of $X$ satisfies $\lim_{n\to \infty}d(x,x_n)=0$ iff $\{n\in \mathbb N: d(x_n,x)\geq r\}$ is finite for every $r>0.$
(3.2). Show that if $\lim_{n\to \infty} d(x,x_n)=0=\lim_{n\to \infty}d(x,x'_n)$ then $\lim_{n\to \infty}d(x_n,x'_n)=0.$
(3.3). For metric space $(X,d)$ and $Y\subset X ,$ define $\overline Y$ to be the set of all ,and only those, $x\in X$ such that $\lim_{n\to \infty}d(x,y_n)=0$ for some sequence $(y_n)_{n\in \mathbb N}$ of members of $Y.$ Prove that $$\overline {(\overline Y)}=\overline Y.$$