PlanetPhysics/Locally Compact Hausdorff Spaces

Definition
A locally compact Hausdorff space $$H_{LC}$$ is a locally compact topological space $$(X_{LC}, \tau)$$ with $$\tau$$ being a Hausdorff topology, 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
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. It would be interesting to have also axiomatic definitions of these two important algebraic topology concepts that are consistent with the T2 axiom.