locally compact Hausdorff spaces

1 Locally compact Hausdorff spaces

Definition 1.1.

A locally compact Hausdorff spacePlanetmathPlanetmath HLC is a locally compact topological space (http://planetmath.org/LocallyCompact) (XLC,τ) with τ being a Hausdorff topology (http://planetmath.org/T2Space), that is, if given any distinct points x,yXLC, there exist disjoint sets U,Vτ such that, UV= (that is, open sets), and with x and y satisfying the conditions that xU and yV.

Remark 1.1.

An important, related concept to the locally compact Hausdorff space is that of a locally compact (topological) groupoidPlanetmathPlanetmathPlanetmathPlanetmath, which is a major concept for realizing extended quantum symmetries in terms of quantum groupoidPlanetmathPlanetmath representations in: quantum algebraic topology (QAT), topological QFT (TQFT), algebraic QFT (AQFT), axiomatic QFT, QCG, and quantum gravity (QG). This has also prompted the relatively recent development of the concepts of homotopyMathworldPlanetmath 2-groupoid and homotopy double groupoid of a Hausdorff space [1, 2]. It would be interesting to have also axiomatic definitions of these two important algebraic topology concepts that are consistent with the T2 axiom.


  • 1 K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff space, Applied Cat. Structures, 8 (2000): 209-234.
  • 2 R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoidPlanetmathPlanetmath of a Hausdorff space, Theory and Applications of Categories 10,(2002): 71-93.
Title locally compact Hausdorff spaces
Canonical name LocallyCompactHausdorffSpaces
Date of creation 2013-03-22 18:19:24
Last modified on 2013-03-22 18:19:24
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 19
Author bci1 (20947)
Entry type Topic
Classification msc 55U40
Classification msc 55-00
Synonym locally compact T2Space
Related topic CharacterizationOfT2Spaces
Related topic T2Space
Related topic LocallyCompact
Related topic LocallyCompactGroupoids
Related topic LocalCompactnessIsHereditaryForLocallyClosedSubspaces
Related topic C_cH_lc
Related topic ExampleOfParacompactTopologicalSpaces
Related topic WeakTopologyOfTheSpaceOfRadonMeasures
Related topic LocallyCompactQuantumGroup
Defines Hausdorff topology
Defines locally compact topological space