Exhaustion by compact sets

In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space $$X$$ is a nested sequence of compact subsets $$K_i$$ of $$X$$ (i.e. $$K_1\subseteq K_2\subseteq K_3\subseteq\cdots$$), such that $$K_i$$ is contained in the interior of $$K_{i+1}$$, i.e. $$K_i\subseteq\text{int}(K_{i+1})$$ for each $$i$$ and $$X=\bigcup_{i=1}^\infty K_i$$. A space admitting an exhaustion by compact sets is called exhaustible by compact sets.

For example, consider $$X=\mathbb{R}^n$$ and the sequence of closed balls $$K_i = \{ x : |x| \le i \}.$$

Occasionally some authors drop the requirement that $$K_i$$ is in the interior of $$K_{i+1}$$, but then the property becomes the same as the space being σ-compact, namely a countable union of compact subsets.

Properties
The following are equivalent for a topological space $$X$$:

(where weakly locally compact means locally compact in the weak sense that each point has a compact neighborhood).
 * 1) $$X$$ is exhaustible by compact sets.
 * 2) $$X$$ is σ-compact and weakly locally compact.
 * 3) $$X$$ is Lindelöf and weakly locally compact.

The hemicompact property is intermediate between exhaustible by compact sets and σ-compact. Every space exhaustible by compact sets is hemicompact and every hemicompact space is σ-compact, but the reverse implications do not hold. For example, the Arens-Fort space and the Appert space are hemicompact, but not exhaustible by compact sets (because not weakly locally compact), and the set $$\Q$$ of rational numbers with the usual topology is σ-compact, but not hemicompact.

Every regular space exhaustible by compact sets is paracompact.