locally compact Hausdorff spaces

1 Locally compact Hausdorff spaces

Definition 1.1.

A locally compact Hausdorff space $H_{LC}$ is alocally compact topological space (http://planetmath.org/LocallyCompact) $(X_{LC},\tau )$ with $\tau$ being aHausdorff topology (http://planetmath.org/T2Space), that is, if given any distinct points $x,y\in X_{LC}$, there exist disjointsets $U,V\in \tau$ 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 interms 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 aHausdorff 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.