The Topology of Metric Spaces

Prove Theorem 2.1.

Theorem 2.1: A set $E$ in a metric space $(M, d)$ is open if and only if its complement, $E^{c} = M \setminus E$, is closed.

Prove Theorem 2.2.

Theorem 2.2:

1. For any collection $\left\{ E_{\alpha }\right\}$ of open sets, $\bigcup_{\alpha } E_{\alpha }$ is open.

2. For any collection $\left\{ F_{\alpha }\right\}$ of closed sets, $\bigcap_{\alpha } F_{\alpha }$ is closed.

3. For any finite collection $\left\{ E_{i}\right\}_{i=1}^{n}$ of open sets, $\bigcap_{i=1}^{n} E_{i}$ is open.

4. For any finite collection $\left\{ F_{i}\right\}_{i=1}^{n}$ of closed sets, $\bigcup_{i=1}^{n} F_{i}$ is closed.

Give examples to show that the word finite cannot be omitted in the last two parts of Theorem 2.2.

Theorem 2.2: (a) For any collection $\left\{ E_{\alpha }\right\}$ of open sets, $\bigcup_{\alpha } E_{\alpha }$ is open. (b) For any collection $\left\{ F_{\alpha }\right\}$ of closed sets, $\bigcap_{\alpha } F_{\alpha }$ is closed. (c) For any finite collection $\left\{ E_{i}\right\}_{i=1}^{n}$ of open sets, $\bigcap_{i=1}^{n} E_{i}$ is open. (d) For any finite collection $\left\{ F_{i}\right\}_{i=1}^{n}$ of closed sets, $\bigcup_{i=1}^{n} F_{i}$ is closed.

Prove Theorem 2.3. Remember that your proof should be in a general metric space.

Theorem 2.3: If $E$ is a compact subset of a metric space, then $E$ is closed.

Is it true that int $(A \cup B)=\operatorname {int}(A) \cup \operatorname {int}(B)$? Prove or give a counterexample. What if union is replaced by intersection?

Let $M$ be any set with the discrete metric. What are the open sets?

Consider $\mathbb {R}$ with the Euclidean metric.

Give an example of a closed and bounded set in a metric space that is not compact.

Consider $\mathbb {R}$ with the Euclidean metric. Give an open cover of $(-10,10]$ that has no finite subcover.

This exercise is about the famous set due to Georg Cantor. If you do not know a definition of the Cantor set, look it up in a real analysis text or ask your professor for a reference.

Prove that a totally bounded set is necessarily bounded. Then, using the discrete metric on any set of your choice, show that a bounded set need not be totally bounded.

Show that if $M$ is a metric space, $A \subseteq M$ is compact, and $U \subseteq M$ is open, then $A \backslash U$ is compact.

Consider metric spaces $M_{1}$ and $M_{2}$.

Complete the proof outlined in Example 2 to show that $C([a, b])$ is separable.

Is $\ell^{1}$ separable? Prove your assertion.

Is $\ell^{\infty }$ separable? Prove your assertion.

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

Let $\left\| \cdot \right\|$ and $\vertiii {\cdot }$ denote two norms defined on the same linear space $X$. Suppose that there exist constants $a>0$ and $b>0$ such that

for each $x \in X$. Show that $(X,\left\| \cdot \right\| )$ is complete if and only if $(X,\vertiii {\cdot })$ is complete. If two norms satisfy (2.1) they are called equivalent norms.

Let $M$ be a set with the discrete metric. Is $M$ complete? Explain.

Define

on $C([a, b])$. Show that this does, indeed, define a norm. Does this norm come from an inner product? Is $\left(C([a, b]),\left\| \cdot \right\|_{1}\right)$ a Banach space?

Show that $\left(C([a, b]),\left\| \cdot \right\|_{2}\right)$ is not complete.

Prove that $\ell^{1}$ is complete.

Prove that $\ell^{\infty }$ is complete.