91Ó°ÊÓ

Show that a convergent sequence in a metric space must be a Cauchy sequence.

Describe the convergent sequences in a metric space \((X, d)\), where \(d\) is the discrete metric.

Let \(S=\\{1 / k: k=1,2,3, \ldots\\}\) and furnish \(S\) with the usual real metric. Answer the following questions about this metric space. (a) Which points are isolated in \(S ?\) (b) Which sets are open and which sets are closed in \(S ?\) (c) Which sets have a nonempty boundary? (d) Which sets are dense in \(S ?\) (e) Which sets are nowhere dense in \(S ?\)

Prove that a subspace of a separable space is separable.

Show that the Hilbert cube is a compact subset of \(\ell_{2}\).

In a metric space \((X, d)\), where \(d\) is the discrete metric, determine which sets are nowhere dense, first category, or residual.

Let \((X, d)\) be a metric space and let \(A\) be a nonempty subset of \(X\). Define \(f: X \rightarrow \mathbb{R}\) by $$ f(x)=\operatorname{dist}(x, A)=\inf \\{d(x, y): y \in A\\} . $$ In Exercise 13.6.11 we established that \(f\) is continuous. Is \(f\) uniformly continuous?

Show that a complete metric space with no isolated points must be uncountable.

A metric space \(X\) is said to be absolutely closed if every isometric image of \(X\) into a space \(Y\) is closed in \(Y\). Show that \(X\) is absolutely closed if and only if it is complete.

Prove that a totally bounded metric space must be separable.