Definition 1.1. A locally compact Hausdorff space H_LC is a locally compact topological space (X_LC,τ) with τ being a Hausdorff topology, that is, if given any distinct points x,y ∈ X_LC, there exist disjoint sets U,V ∈ τ such that, U ∩ V = ∅ (that is, open sets), and with x and y satisfying the conditions that x ∈ U and y ∈ 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.