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, $U\cap V=\emptyset$ (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.

References

• 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 groupoid 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