locally compact Hausdorff spaces

1 Locally compact Hausdorff spaces

Definition 1.1.

A locally compact Hausdorff space $H_{LC}$ is a locally compact topological space (http://planetmath.org/LocallyCompact) $(X_{LC},\tau )$ with $\tau$ being a Hausdorff topology (http://planetmath.org/T2Space), that is, if given any distinct points $x,y\in X_{LC}$, there exist disjoint sets $U,V\in \tau$ such that, $U\cap V=\mathrm{\varnothing }$ (that is, open sets), and with $x$ and $y$ satisfying the conditions that $x\in U$ and $y\in V$.

Remark 1.1.

An important, related concept to the locally compact Hausdorff space is that of a locally compact (topological) groupoid, which is a major concept for realizing extended quantum symmetries in terms of quantum groupoid 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 homotopy 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.

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