Interesting Proofs about Metric Spaces?

Question

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!

Answer 1

$1)$Prove or disprove with a counterxample: Is a countable intersection of open sets always open?

$2)$Prove that a finite intersection of open sets is open.

$3)$Let the space $C[0,1]=\{f[0,1] \rightarrow \mathbb{R}|f$ continuous on $[0,1]\}$ and $d(f,g)= \int_0^1|f(x)-g(x)|dx$. Prove that $d$ is a metric.

$4)$Let (X,d) be a metric space.Prove that the collection of sets $T=\{A \subseteq X| \forall x \in A,\exists \epsilon>0$such that $B(x, \epsilon) \subseteq A\}$ is a topology on $X$.You need only to look the definition of a topolgy to solve this.

$5)$ Prove that the set of rational numbers is not an open subset of $\mathbb{R}$ under the metric $d(x,y)=|x-y|$(usual metric)

$6)$Prove that the set $A=\{(x,y) \in \mathbb{R}^2|x+y>1\}$ is an open set in $\mathbb{R}^2$ under the metric $d((x_1,y_1),(x_2,y_2))=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

$7)$Let $(X,d)$ be a metric space and $A \subset X$.We define $(x_0,A)=\inf\{d(x_0,y)|y \in A \}$. This quantity is called the distance between $x_0$ and $A$.Prove that the function $f:X \rightarrow \mathbb{R}$ such that $f(x)=d(x,A)$ is lipschitz continuous.

$8)$A set $A$ in a metric (and topological in general)space is closed if $X$ \ $A$ is open. Prove that the set $\mathbb{Z}$ is a closed subsets of the real line under the usual metric.Also prove that the set of rational numbers in not closed under the same metric.

$9)$A subset $Y$ of metric space X is connected if there DO NOT exist two open sets $A,B \subseteq X$ such that $Y=A \cup B$ and $A \cap B= \emptyset$. Prove that the set $(0,1)$ is a connected subset of $ \mathbb{R}$ under the usual metric.Also prove that $\mathbb{Q}$ is not connected in $\mathbb{R}$ under the usual metric.

$10)$Firstly prove that an interval $(a,b),(a, + \infty),(- \infty,a)$($0<a<b$) are open sets in $\mathbb{R}$ under the usual metric ($d(x,y)=|x-y|$)

Secondly prove that the set $[a,b]$ is closed in $\mathbb{R}$(use the definition of a previous exercise and the first part of the exercise)

Finally, prove that $\bigcup_{n=1}^\infty [1+\frac{1}{n},2-\frac{1}{n}]=(1,2)$

This is an example in which an infinite union of closed sets in a metric space need not to be a closed set.

$11)$Let $(X,d)$ be a metric space .We define the diameter of a set $A$ as $diam(A)=\sup \{d(x,y)|x,y \in A\}$.Suppose that $B$ is a bounded subset of X and $C \subseteq B$.Prove that $diam(C) \leqslant diam(B)$

$12)$Let $X$ be the space of continuous functions on $[0, 1]$($C[0,1]$) with the metric $d(f,g)= \sup_{x \in [0,1]}|f(x)-g(x)|$.Show that $d$ is indeed a metric. Also show that the subset $A = \{f ∈ X | f(x) > 1,$ for $x \in [1/3, 2/3]\}$ is open in $X$.

$13)$Let $(X,d)$ be a metric space.Define $A+B=\{x+y|x \in A ,y \in B \}$ and $x+A=\{x+y| y \in A\}$ where $A,B \subseteq X$.Prove that if $A,B$ are open sets then $A+B,x+A$ are also open sets.

$14)$Let $(X,d)$ be a metric space.A sequence $x_n \in X$ converges to $x$ if $\forall \epsilon >0 ,\exists n_0 \in \mathbb{N}$ such that $d(x_n,x)< \epsilon, \forall n \geqslant n_0$.Consider the space $(\mathbb{R}^m,d)$ with the euclideian metric.Prove that $x_n \rightarrow x=(x_1,x_2...x_m)$ in $\mathbb{R}^m$ if and only if $x_n^j \rightarrow x_j \in \mathbb{R}, \forall j \in \{1,2...m\}$(A sequence in $\mathbb{R}^m$ has the form $x_n=(x_n^1,x_n^2...x_n^m))$

$15)$Let a function $f:(X,d_1) \rightarrow (Y,d_2)$.Prove that $f$ is continuous in $X$ if and only if for every sequence $x_n \rightarrow x$ in $X$ we have $f(x_n) \rightarrow f(x)$ in $Y$.

I hope this helps you a bit.

Answer 2

Let $(X,d)$ be a metric space.

• If $f:(X,d)\to (X,d)$ is continuous and $f\circ f=f$ then $f(X)$ is closed.
• Every sequence in $(X,d)$ converges to at most one point in $X$.
• For each $n\in\mathbb{N}$, there exists a metric $\rho$ on $X$ such that for each $x,y\in X, \rho(x,y)\leq n$ and the family of open balls in $(X,d)$ coincides with the family of open balls in $(X,\rho)$.
• If $X=\mathbb{R}$ and $d$ is the usual metric then every closed interval (or in fact any closed set) is the intersection of a family of open sets, i.e. a $G_\delta$ set.
• Every point of $X$ has a countable neighborhood base, i.e. for each $x\in X,$ there exists a countable family $\eta(x)$ of open sets such that for any open neighborhood $U$ of $x$, there exists $V\in \eta(x)$ such that $x\in U\subseteq V$.
• If $(X,d)$ is second countable, i.e. if there exists a countable family $\mathcal{B}$ of open sets in $(X,d)$ such that for each open set $U$ in $X$, there exists an open set $V\in \mathcal{B}$ such that $V\subseteq U$, then $(X,d)$ is first countable but the converse is not necessarily true.
• If $E,F$ are two disjoint closed subsets of $X$ then there exist disjoint $U,V$ open sets in $(X,d)$ such that $E\subseteq U,\ F\subseteq V$ and $U\cap V=\emptyset$.
• If $a\in X$ and $F$ is a closed subset of $X$ with $x\notin F$ then there exists $U, V$ open subsets of $X$ such that $x\in U,\ F\subseteq V$ and $U\cap V=\emptyset$.
• If $X=\mathbb{R}$ and $d$ is the usual metric then every open subset of $X$ is at most a countable union of disjoint open intervals.

Answer 3

The "discrete metric" on a space $X$ is one in which $d(x, y) = 1$ if $x \ne y$, and $d(x, x) = 0$.

(a) Show that for any set $X$, the discrete metric on $X$ is, in fact, a metric.

(b) Show that every function from $X$ with its discrete metric to any metric space $Y$ is in fact continuous.

(c) Show that a continuous function from any metric space $Y$ to the space $X$ (with its discrete metric) must be constant.

Answer 4

(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 $$\overline {(\overline Y)}=\overline Y.$$

Answer 5

• Every convergent sequence is bounded.
• The closure of an open ball $B(a;\delta)$ is a subset of the closed ball centered at $a$ with radius $\delta$.
• The union of a sequence of closed subsets doesn't have to be closed.

Answer 6

A lot of really good metric problems have already been posted, but I'd like to add that you may want to try Topology Without Tears by Sidney A. Morris. The book is extremely rigorous and has hundreds of problems at varying difficulties; as with a lot of proofs, some take seconds, some might take you days.