Possible Generalizations of The Heine-Borel Theorem

Solution 1:

For $(1)$, consider the discrete metric (all distinct points are distance $1$ apart) on any infinite set. This is (trivially) complete and bounded, but not compact.

For $(2)$, the following are equivalent for a metric space $X$:

• $X$ has the Heine-Borel property.

• For every point $x\in X$, every closed ball of finite radius centered on $x$ is compact.

• For some point $x\in X$, every closed ball of finite radius centered on $x$ is compact.

The implications $1\rightarrow 2\rightarrow 3$ are immediate; the interesting one is $3\rightarrow 1$. Suppose $a\in X$ has the property that every closed ball of finite radius centered on $a$ is compact, and let $S\subseteq X$ be closed and bounded; we want to show that $S$ is compact. Let $r=\sup\{d(a, v): v\in S\}$; then $S$ is a closed subset of the closed ball centered on $a$ of radius $r+1$. But any closed subset of a compact set is compact, and that closed ball is compact by assumption on $a$.

Note that this tells us that every space with the Heine-Borel property is $\sigma$-compact, that is, a union of countably many compact subsets. The converse is false: any countably infinite discrete metric space is trivially $\sigma$-compact, but doesn't have the Heine-Borel property (since it itself is closed-in-itself, bounded, but not compact).