Compactness

Prove that a Cauchy sequence in a metric space is bounded.

Prove that every totally bounded metric space is bounded.

Prove that every bounded subset of $\mathbb {R}^{n}$ is totally bounded.

Prove that the closure of a totally bounded set in a metric space is totally bounded.

Prove that a metric space ( $M, d$ ) is totally bounded if and only if every sequence in $M$ has a Cauchy subsequence.

Prove that the closure of a bounded set in a metric space is bounded.

Prove that a compact set in a metric space is closed.

Prove that a closed subset of a compact metric space is compact.

Prove that a metric space ( $M, d$ ) is compact if and only if every infinite subset of $M$ has a cluster point in $M$.

Prove that a subset of a complete metric space $M$ is compact if and only if it is totally bounded and closed.

In the metric space $(\mathbb {Q}, d)$, where $d(x, y)=\left|x-y\right|$, determine all subspaces that are:

Prove or disprove: the set $\mathbb {N}$ is a compact subset of ( $\mathbb {R}, d$ ), where $d(x, y)=\left|x-y\right|$.

Let ( $M, d$ ) be a complete metric space. Prove that a set $A \subset M$ is totally bounded if and only if $\bar{A}$ is compact.

Let ( $M, d$ ) be the metric space of all bounded sequences, with metric $d(a, b)=\sup \left\{ \left|a_{n}-b_{n}\right|: n \in \mathbb {N}\right\}$, where $a=\left\{ a_{n}\right\}$ and $b=\left\{ b_{n}\right\}$. (See Exercise 2.1.13.) Let $B$ be the closed ball of radius 1 centered at the zero sequence $(0,0, \ldots )$. Prove that $B$ is bounded and closed but not sequentially compact.

Let ( $M, d$ ) be a metric space and let $A \subset M$ be a set such that $A \cap K$ is compact for every compact set $K \subset M$. Prove that $A$ is closed in $M$.

Let ( $M, d$ ) be a metric space. Prove that the distance between a closed set and a compact set that are disjoint cannot be 0 . Give an example to show that this is false if both sets are only closed.

Let $K_{1}, K_{2}$ be disjoint compact subsets of a metric space ( $M, d$ ). Prove that there exist disjoint, open sets $G_{1}, G_{2}$ such that $K_{1} \subset G_{1}$ and $K_{2} \subset G_{2}$.

Prove that a metric space $M$ is compact if and only if every collection of closed subsets of $M$ with the finite intersection property (meaning that every finite subcollection has non-empty intersection) has a non-empty intersection.

Prove that a metric space $M$ is compact if and only if every decreasing sequence of nonempty closed sets in $M$ has nonempty intersection.

Prove that the union of a finite number of compact sets in a metric space is compact.

Let $M$ be a compact metric space and let $\left\{ G_{n}\right\}_{n=1}^{\infty }$ be an increasing sequence of open subsets of $M$ such that $M=\cup_{n=1}^{\infty } G_{n}$. Prove that $M=G_{k}$ for some $k \in \mathbb {N}$.

Let $\left\{ M_{i}\right\}_{i=1}^{n}$ be compact metric spaces. Prove that $M:=M_{1} \times M_{2} \times \cdots \times M_{n}$ is a compact metric space with respect to metric $d$ as in Exercise 2.1.14.

Determine if $C[0,1]$ is sequentially compact.

Determine if the metric space ( $M, d$ ) is sequentially compact, where $d(x, y)=0$ if $x=y$ and $d(x, y)=1$ if $x \neq y$.

Determine if the metric space ( $\mathbb {R}, d$ ) is sequentially compact, where

Determine if the metric space ( $C[0,1], d$ ) is sequentially compact, where

Determine if the metric space $(C(-\infty , \infty ), d)$ is sequentially compact, where

Prove Theorem 2.3.18.

Theorem 2.3.18: Let $(M,d)$ be a compact metric space and let $f: M \rightarrow \mathbb {R}$ be continuous. Then $f$ is bounded and it attains its extreme values on $M$.

Let ( $M, d$ ) be a metric space and let $f, g: M \rightarrow \mathbb {R}$ be continuous. Prove that $\max \left\{ f, g\right\}$ and $\min \left\{ f, g\right\}$ are continuous.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$, let $x_{0} \in \mathbb {R}$, and suppose that for every interval $(a, b)$ that contains $x_{0}$, the function $f$ is unbounded on ( $a, b$ ). Prove that $f$ is not continuous at $x_{0}$.

Let ( $M, d$ ) be a metric space, let $f: M \rightarrow \mathbb {R}$ be continuous, and let $E \subset M$. Prove that $\sup \left\{ \left|f(x)\right|: x \in E\right\} =\sup \left\{ \left|f(x)\right|: x \in \bar{E}\right\}$.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be a strictly increasing continuous function. Prove that $f^{-1}$ is continuous.

Let ( $M, d$ ) be a metric space, let $f, g: M \rightarrow \mathbb {R}$ be continuous, and let $\alpha \in \mathbb {R}$. Prove that $\alpha f, f+g$ and $f g$ are continuous functions. Moreover, if $g(x) \neq 0$ on $M$, prove that $f / g$ is continuous.

Let $f$ be a real-valued function on a metric space $X$. Show that the set of points at which $f$ is continuous is the intersection of a countable collection of open sets. Conclude that there is no real-valued function on $\mathbb {R}$ that is continuous at the rational numbers only.

Let ( $M, d$ ) be a metric space and $K$ a compact subset of $M$. Suppose that a function $f: K \rightarrow \mathbb {R}$ is continuous. Prove that $f(K)$ is a compact subset of $\mathbb {R}$.

Let ( $M, d$ ) be a metric space and $A \subset M$. Suppose that functions $f, g: M \rightarrow \mathbb {R}$ are continuous and $f(x)=g(x)$ when $x \in A$. Prove that $f(x)=g(x)$ for all $x \in \bar{A}$.

Let ( $M, d$ ) be a metric space, let $A \subset M$, and let $h(x)=d(x, A)= \inf \left\{ d(x, a): a \in A\right\}$. Prove that $d$ is a continuous function on $M$.

Let ( $M, d$ ) be a metric space and let $A$ and $B$ be disjoint, closed subsets of $M$. Prove that there exists a continuous, real-valued function $f$ on $M$ such that $0 \leq f(x) \leq 1$, for all $x \in M$, $f(x)=0$ on $A$, and $f(x)=1$ on $B$.

Let $\left\{ f_{n}\right\}$ be a sequence of continuous functions that converge to $f$ uniformly on $[a, b]$ and let $\left\{ x_{n}\right\}$ be a sequence in $[a, b]$ that converges to $x$. Prove that $\left\{ f_{n}\left(x_{n}\right)\right\}$ converges to $f(x)$.

Prove Theorem 2.3.17.

Theorem 2.3.17: Let $(M,d)$ be a metric space. Every uniformly continuous map $f: M_{1} \rightarrow M_{2}$ is continuous. If, in addition, $M_{1}$ is compact then every continuous map $f: M_{1} \rightarrow M_{2}$ is uniformly continuous.

Determine whether the function $F$ in Example 2.3.12 is uniformly continuous on $C[0,1]$.

Example 2.3.12: Let $a \in [0,1]$. The function $F: C[0,1] \rightarrow \mathbb {R}$, defined by $F(f) = f(a)$, is continuous on the metric space $C[0,1]$.

Determine whether the function $F$ in Example 2.3.12 is continuous on ( $C[0,1], d$ ) with $d$ defined as in Exercise 2.1.3, by

Is it uniformly continuous?

Example 2.3.12: Let $a \in [0,1]$. The function $F: C[0,1] \rightarrow \mathbb {R}$, defined by $F(f)=f(a)$, is continuous on the metric space $C[0,1]$ with the metric $d(f, g)=\sup \left\{ \left|f(x)-g(x)\right|: x \in [0,1]\right\}$.

Let ( $M, d$ ) be a metric space and $\left\{ x_{n}\right\} \subset A \subset M$ a Cauchy sequence. Suppose that a function $f: M \rightarrow \mathbb {R}$ is uniformly continuous on $A$. Prove that $\left\{ f\left(x_{n}\right)\right\}$ is a Cauchy sequence in $\mathbb {R}$. What if $f$ is assumed to be only continuous?

Let ( $M, d$ ) be a metric space and let $f: M \rightarrow \mathbb {R}$. Prove that the following statements are equivalent:

Prove that the function $f=\chi_{(0,1]}$ is upper semicontinuous at every point except at $x=0$ and it is lower semicontinuous at every point except at $x=1$.

Let ( $M, d$ ) be a metric space. Suppose that, for every $\varepsilon >0$, there exists $\delta >0$, a metric space ( $W, \rho$ ), and a mapping $\Phi : M \rightarrow W$ so that $\Phi (M)$ is totally bounded, and whenever $x, y \in M$ are such that $\rho (\Phi (x), \Phi (y))<\delta$, then $d(x, y)<\varepsilon$. Prove that $M$ is totally bounded.

Prove Banach Fixed-point Theorem (Theorem 2.3.21).

Theorem 2.3.21 (Banach Fixed-point Theorem): A contraction on a metric space has exactly one fixed point, i.e., the point $x$ such that $f(x)=x$.