Math Problems Vault

Problem 21.22

Show that Euclidean space $\mathbb {R}^{n}$ is locally compact.

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Problem 21.23

Show that $\ell^{p}$ , for $1 \leq p \leq \infty$ , fails to be locally compact.

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Problem 21.24

Show that $C([0,1])$ , with the topology induced by the maximum norm, is not locally compact.

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Problem 21.25

(i)(ii)(iii)

Let $\rho : X \times X \rightarrow \mathbb {R}$ be a metric on a set $X$ . For $A \subseteq X$ , consider the distance function

\operatorname {dist}_{A}: X \rightarrow [0, \infty ) .

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(i)

Show that the function $\operatorname {dist}_{A}$ is continuous.

(ii)

If $A \subseteq X$ is closed and $x$ is a point in $X$ , show that $\operatorname {dist}_{A}(x) = 0$ if and only if $x \in A$ .

(iii)

If $A \subseteq X$ is closed and $x \in X$ , show that there may not exist a point $x_{0}$ in $A$ for which $\operatorname {dist}_{A}(x) = \rho \left(x, x_{0}\right)$ , but there is such a point $x_{0}$ if $K$ is compact.

Problem 21.26

Show that property (ii) in the statement of Theorem 7 is equivalent to the assertion that two disjoint closed subsets of $X$ , one of which is compact, may be separated by disjoint neighborhoods.

Theorem 7: Let $(X, \mathcal{T})$ be a Hausdorff space. Then the following four properties are equivalent: (i) $(X, \mathcal{T})$ is locally compact; (ii) if $O$ is a neighborhood of a compact subset $K$ of $X$ , then there is a neighborhood $U$ of $K$ that has compact closure contained in $O$ ; (iii) if $O$ is a neighborhood of a compact subset $K$ of $X$ , then the constant function on $K$ that takes the value $1$ may be extended to a function $f$ in $C_{c}(X)$ for which $0 \leq f \leq 1$ on $X$ and $f$ vanishes outside of $O$ ; (iv) for $K$ a compact subset of $X$ and $\mathcal{F}$ a finite open cover of $K$ , there is a partition of unity subordinate to $\mathcal{F}$ consisting of functions of compact support.

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