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 $X=\bigcup _{i=1}^{\mathrm{\infty }}{K}_i$. 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 $X$:

1. $X$ is exhaustible by compact sets.
2. $X$ is σ-compact and weakly locally compact.
3. $X$ is Lindelöf and weakly locally compact.

(where weakly locally compact means locally compact in the weak sense that each point has a compact neighborhood).

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 $\mathbb{\mathbb{Q}}$ of rational numbers with the usual topology is σ-compact, but not hemicompact.

Every regular space exhaustible by compact sets is paracompact.

References

• Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. ISBN:0-8218-1221-1.
• Hans Grauert and Reinhold Remmert, Theory of Stein Spaces, Springer Verlag (Classics in Mathematics), 2004. ISBN:978-3540003731.
• Lee, John M. (2011). Introduction to topological manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-7939-1. 

External links

• "Exhaustion by compact sets". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}. 
• "Existence of exhaustion by compact sets". https://math.stackexchange.com/questions/1360900.