Compact Metric Spaces

Consider the metric space $\mathbb {Q}$ consisting of the rational numbers with the metric induced by the absolute value. Which subspaces of $\mathbb {Q}$ are complete and which are compact?

Let $B = B(x, r)$ be an open ball in Euclidean space $\mathbb {R}^{n}$. Show that $B$ fails to be compact by

When is a set $X$ with the discrete metric a compact metric space?

Prove Proposition 23.

Proposition 23: If a subspace $K$ of a metric space $X$ is compact, then $K$ is a closed, bounded subset of $X$.

Prove that if $X$ is compact, then $K \subseteq X$ is compact if and only if it is closed subset of $X$.

For the countable Cartesian product of metric spaces considered in Proposition 7, show that the product is compact if and only if each factor space is compact.

Proposition 7: Let $X = \prod_{n=1}^{\infty } X_{n}$ be the Cartesian product of the countable collection of metric spaces $\left\{ (X_{n}, \rho_{n})\right\}_{n=1}^{\infty }$. For points $u = \left\{ u_{n}\right\}$ and $v = \left\{ v_{n}\right\}$ in $X$, define $\rho (u, v) = \sum_{n=1}^{\infty } \frac{1}{2^{n}} \cdot \frac{\rho_{n}(u_{n}, v_{n})}{1 + \rho_{n}(u_{n}, v_{n})}$. Then $\rho$ is a metric on $X$ for which sequential convergence is componentwise convergence in each component metric space $(X_{n}, \rho_{n})$.

Let $E$ be a subset of the compact metric space $X$. Show the metric subspace $E$ is compact if and only if $E$ is a closed subset of $X$.

Let $\left\{ F_{n}\right\}_{n = 1}^{\infty }$ be a descending countable collection of non-empty closed subsets of a compact metric space $X$. Show that $\bigcap_{n = 1}^{\infty } F_{n} \neq \emptyset$.

For a subset $E$ of a metric space $X$, show that $E$ is totally bounded if and only if its closure $\bar{E}$ is totally bounded.

For a subset $E$ of a complete metric space $X$, show that $E$ is totally bounded if and only if its closure $\bar{E}$ is compact.

Let $B = \left\{ \left\{ x_{n}\right\} \in \ell^{2} \mid \sum_{n = 1}^{\infty } x_{n}^{2} \leq 1\right\}$ be the closed unit ball in $\ell^{2}$. Show that $B$ fails to be compact by

Let $B = \left\{ f \in L^{2}[a, b] \mid \left\| f\right\|_{2} \leq 1\right\}$ be the closed unit ball in $L^{2}[a, b]$. Show that $B$ fails to be compact by

Let $X$ be a totally bounded metric space.

Let $\rho$ be a metric on a set $X$. Define

Verify that $\tau$ is a bounded metric on $X$ and convergence of sequences with respect to the $\rho$ metric and the $\tau$ metric is the same. Conclude that sets that are closed with respect to the $\rho$ metric are closed with respect to the $\tau$ metric and that sets that are open with respect to the $\rho$ metric are open with respect to the $\tau$ metric.

Let $E$ be a subset of Euclidean space $\mathbb {R}^{n}$. Assume that every continuous real-valued function on $E$ takes a minimum value. Is $E$ necessarily closed and bounded?

Suppose $f$ is a continuous real-valued function on Euclidean space $\mathbb {R}^{n}$ with the property that there is a positive number $c$ such that $\left|f(x)\right| \geq c \cdot \left\| x\right\|$ for all $x \in \mathbb {R}^{n}$. Show that if $K$ is a compact set of real numbers, then its inverse image under $f, f^{-1}(K)$, also is compact. (Mappings with this property are called proper.)

For a compact metric space ($X, \rho$), show that there are points $u, v \in X$ for which $\rho (u, v) = \operatorname {diam} X$.

Let $K$ be a compact subset of the metric space ($X, \rho$) and $x_{0}$ belong to $X$. Show that there is a point $z \in K$ for which

Let $K$ be a compact subset of the metric space $X$. For a point $x \in X \sim K$, show that there is an open set $\mathcal{U}$ containing $K$ and an open set $\mathcal{O}$ containing $x$ for which $\mathcal{U} \cap \mathcal{O} = \emptyset$.

Let $A$ and $B$ be subsets of a metric space ($X, \rho$). Define

If $A$ is compact and $B$ is closed, show that $A \cap B = \emptyset$ if and only if $\operatorname {dist}(A, B) > 0$.

Let $K$ be a compact subset of a metric space $X$ and $\mathcal{O}$ an open set containing $K$. Use the preceding problem to show that there is an open set $\mathcal{U}$ for which $K \subseteq \mathcal{U} \subseteq \overline{\mathcal{U}} \subseteq \mathcal{O}$.