Munkres Topology Solution: Chapter 7

§43 Complete Metric Spaces

Ex.43.1

Let X be a metric space.
(a) Suppose that for some $\epsilon>0$, every $\epsilon$-ball in $X$ has compact closure. Show that $X$ is complete.
(b) Suppose that for each $x\in X$ there is an $\epsilon>0$ such that the ball $B(x,\epsilon)$ has compact closure. Show by means of an example that $X$ need not be complete.