hemicompact space

A topological spaceMathworldPlanetmath (X,τ) is called a hemicompact space if there is an admissible sequence in X, i.e. there is a sequence of compact sets (Kn)n in X such that for every KX compact there is an n with KKn.

PropositionPlanetmathPlanetmath. Let (X,τ) be a first countable hemicompact space. Then X is locally compact.


Let KnKn+1 be an admissible sequence of X. Assume for contradictionMathworldPlanetmathPlanetmath that there is an xX without compact neighborhoodMathworldPlanetmathPlanetmath. Let UnUn+1 be a countable basis for the neighbourhoods of x. For every n choose a point xnUnKn. The set K:={xn:n}{x} is compact but there is no n with KKn. We have a contradiction. ∎

Proposition. Let (X,τ) be a locally compact and σ-compact space. Then X is hemicompact.


By local compactness we choose a cover XiIUi of open sets with compact closure (take a compact neighborhood of every point). By σ-compactness there is a sequence (Kn)n of compacts such that X=nKn. To each Kn there is a finite subfamily of (Ui)iI which covers Kn. Denote the union of this finite family by Un for each n. Set K~n:=k=1nUk¯. Then (K~n)n is a sequence of compacts. Let KX be compact then there is a finite subfamily of (Ui)iI covering K. Therefore KKn for some n. ∎

Title hemicompact space
Canonical name HemicompactSpace
Date of creation 2013-03-22 19:08:18
Last modified on 2013-03-22 19:08:18
Owner karstenb (16623)
Last modified by karstenb (16623)
Numerical id 8
Author karstenb (16623)
Entry type Definition
Classification msc 54-00
Related topic SigmaCompact
Defines hemicompact space