Interesting Proofs about Metric Spaces?

$1)$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.

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.

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.