PlanetPhysics/Locally Compact Hausdorff Spaces

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, $U\cap V=\mathrm{\varnothing }$ (that is, open sets), and with $x$ and $y$ satisfying the conditions that $x\in U$ and $y\in V$.

Remark

An important, related [[../PreciseIdea/|concept]] to the locally compact Hausdorff space is that of a locally compact ([[../CoIntersections/|topological]]) [[../EquivalenceRelation/|groupoid]], which is a major concept for realizing [[../TopologicalOrder2/|extended quantum symmetries]] in terms of [[../WeakHopfAlgebra/|quantum groupoid]] [[../CategoricalGroupRepresentation/|representations]] in: [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]] ([[../QuantumOperatorAlgebra5/|QAT]]), topological QFT ([[../SUSY2/|TQFT]]), [[../CoIntersections/|algebraic]] QFT ([[../SUSY2/|AQFT]]), [[../PureState/|axiomatic QFT]], [[../QuantumCompactGroupoids/|QCG]], and [[../LQG2/|quantum gravity]] ([[../SUSY2/|QG]]). This has also prompted the relatively recent development of the concepts of [[../ThinEquivalence/|homotopy]] [[../InfinityGroupoid/|2-groupoid]] and homotopy double groupoid of a Hausdorff space . It would be interesting to have also axiomatic definitions of these two important [[../CubicalHigherHomotopyGroupoid/|algebraic topology]] concepts that are consistent with the T2 axiom.