locally compact Hausdorff spaces
1 Locally compact Hausdorff spaces
Definition 1.1.
A locally compact Hausdorff space^{} ${H}_{LC}$ is a locally compact topological space (http://planetmath.org/LocallyCompact) $({X}_{LC},\tau )$ with $\tau $ being a Hausdorff topology (http://planetmath.org/T2Space), 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 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.
References
- 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.
Title | locally compact Hausdorff spaces |
Canonical name | LocallyCompactHausdorffSpaces |
Date of creation | 2013-03-22 18:19:24 |
Last modified on | 2013-03-22 18:19:24 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 19 |
Author | bci1 (20947) |
Entry type | Topic |
Classification | msc 55U40 |
Classification | msc 55-00 |
Synonym | locally compact T2Space |
Related topic | CharacterizationOfT2Spaces |
Related topic | T2Space |
Related topic | LocallyCompact |
Related topic | LocallyCompactGroupoids |
Related topic | LocalCompactnessIsHereditaryForLocallyClosedSubspaces |
Related topic | C_cH_lc |
Related topic | ExampleOfParacompactTopologicalSpaces |
Related topic | WeakTopologyOfTheSpaceOfRadonMeasures |
Related topic | LocallyCompactQuantumGroup |
Defines | Hausdorff topology |
Defines | locally compact topological space |