nLab hemicompact space

Definition

A topological space $X$ is called hemicompact iff there exists a sequence $(K_n)_{n\in\mathbb{N}}$ of compact subsets $K_n\subseteq X$ such that $X=\bigcup_{n\in\mathbb{N}} K_n$ and $K_n \subseteq int(K_{n+1})$ holds.

Properties

If $X$ is hemicompact and $(K_n)$ a sequence as in the definition, then every compact subset $K\subseteq X$ is contained in one of the $K_n$. In other words: $\lbrace K_n \mid n\in\mathbb{N}\rbrace$ is cofinal in the set of all compact subsets (partially ordered by inclusion)

Proposition

Locally compact (every point has a compact neighborhood) and $\sigma$-compact spaces are hemicompact (and also paracompact).

Proposition

A first-countable and hemicompact space is locally compact (every point has a compact neighborhood).

Here the condition to be first-countable can’t be dropped as there are hemicompact spaces, which are not locally compact. An example is the Arens-Fort space as seen in Joshi 83, Chapter 4, Section 2, Example 10, which is not even compact.

Proposition

A hemicompact space is $\sigma$-compact.

Example

The field of rational numbers $\mathbb{Q}$ is $\sigma$-compact, but not hemicompact.

(Willard 04, p. 126, answer by Eric Wofsey on MSE/4209303)

References

• Stephen Willard. General Topology (2004). Dover Publications. ISBN 0-486-43479-6. pdf
• K. D. Joshi. Introduction to General Topology (1983). New Age International. ISBN 978-0-47027-556-6. pdf

Related concepts

• orthocompact space

References

See also:

• Wikipedia, Hemicompact space
• π-Base, Hemicompact