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locally compact Hausdorff spaces (Topic)

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.

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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Keywords:  locally compact Hausdorff spaces

Cross-references: algebraic topology, 2-groupoid, homotopy, QG, quantum gravity, QCG, axiomatic QFT, AQFT, algebraic, TQFT, QFT, QAT, quantum algebraic topology, representations, quantum groupoid, extended quantum symmetries, groupoid, topological, concept
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This is version 1 of locally compact Hausdorff spaces, born on 2009-03-15.
Object id is 597, canonical name is LocallyCompactHausdorffSpaces.
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Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
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