nLab hemicompact space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

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

Properties

If XX is hemicompact and (K n)(K_n) a sequence as in the definition, then every compact subset KXK\subseteq X is contained in one of the K nK_n. In other words: {K nn}\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 σ\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 σ\sigma-compact.

Example

The field of rational numbers \mathbb{Q} is σ\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

References

See also:

Last revised on June 10, 2024 at 11:24:08. See the history of this page for a list of all contributions to it.