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%%% Primary Title: locally compact Hausdorff spaces
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%%% Filename: LocallyCompactHausdorffSpaces.tex
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\begin{document}
\section{Locally compact Hausdorff spaces}
\begin{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, $U\cap V=\emptyset$ (that is, open sets), and with $x$ and $y$ satisfying the conditions that $x \in U$ and $y \in V$.
\end{definition}
\begin{remark}
An important, related \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} to the locally compact Hausdorff space is that of a \emph{locally compact (\htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html})
\htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/EquivalenceRelation.html}}, which is a major concept for realizing \htmladdnormallink{extended quantum symmetries}{http://planetphysics.us/encyclopedia/TopologicalOrder2.html} in
terms of \emph{\htmladdnormallink{quantum groupoid}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html} \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html}} in: \htmladdnormallink{Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html} (\htmladdnormallink{QAT}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}), topological QFT (\htmladdnormallink{TQFT}{http://planetphysics.us/encyclopedia/SUSY2.html}), \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} QFT (\htmladdnormallink{AQFT}{http://planetphysics.us/encyclopedia/SUSY2.html}), \htmladdnormallink{axiomatic QFT}{http://planetphysics.us/encyclopedia/PureState.html}, \htmladdnormallink{QCG}{http://planetphysics.us/encyclopedia/QuantumCompactGroupoids.html}, and \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html} (\htmladdnormallink{QG}{http://planetphysics.us/encyclopedia/SUSY2.html}). This has also prompted the relatively recent development of the concepts of \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} \htmladdnormallink{2-groupoid}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} and \textbf{homotopy \emph{double} groupoid} of a
Hausdorff space \cite{HKK, BHKP}. It would be interesting to have also axiomatic definitions of these two important \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} concepts that are consistent with the T2 axiom.
\end{remark}
\begin{thebibliography}{9}
\bibitem{HKK}
K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff space,
\emph{Applied Cat. Structures}, \textbf{8} (2000): 209-234.
\bibitem{BHKP}
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} \textbf{10},(2002): 71-93.
\end{thebibliography}
\end{document}
