Locally compact Hausdorff spaces

Definition 1.1   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 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.