PlanetPhysics/Locally Compact Hausdorff Spaces

Locally compact Hausdorff spaces

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Definition

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A locally compact Hausdorff space   is a locally compact topological space   with   being a Hausdorff topology, that is, if given any distinct points  , there exist disjoint sets   such that,   (that is, open sets), and with   and   satisfying the conditions that   and  .

Remark

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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.

All Sources

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[1] [2]

References

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  1. 1.0 1.1 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. 2.0 2.1 R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff space, {\it Theory and Applications of Categories} 10 ,(2002): 71-93.