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locally compact Hausdorff space (Topic)
Definition 0.1   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$.

Remarks An important, related concept to the locally compact Hausdorff space is that of a locally compact (topological) groupoid, which is a major contender 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. 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.

Bibliography

1
K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff space, Applied Cat. Structures, 8 (2000): 209-234.
2
R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff space, Theory and Applications of Categories 10,(2002): 71-93.



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See Also: characterization of $T2$ spaces, Hausdorff space, locally compact, locally compact groupoid, local compactness is hereditary for locally closed subspaces, $C_c (H_{lc})$, example of paracompact topological spaces, weak-* topology of the space of Radon measures

Other names:  locally compact T2Space
Also defines:  Hausdorff topology, locally compact topological space
Keywords:  T2Space, LocallyCompact, LocallyCompactGroupoid, Hausdorff topology
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Cross-references: T2 axiom, consistent, topology, definitions, Hausdorff space, homotopy, development, Quantum Gravity, axiomatic, algebraic, TQFT, QFT, QAT, quantum algebraic topology, representations, quantum groupoid, terms, extended quantum symmetries, groupoid, locally compact, open sets, disjoint, points
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This is version 11 of locally compact Hausdorff space, born on 2008-08-20, modified 2008-11-23.
Object id is 10954, canonical name is LocallyCompactHausdorffSpace.
Accessed 667 times total.

Classification:
AMS MSC: 55-00 (Algebraic topology :: General reference works )
 55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory)
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