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locally compact Hausdorff spaces

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\begin{document}
\section{Locally compact Hausdorff spaces}

\begin{definition}  A locally compact Hausdorff space $H_{LC}$ is a 
\PMlinkname{locally compact topological space}{LocallyCompact} $(X_{LC}, \tau)$ with $\tau$ being a
\PMlinkname{Hausdorff topology}{T2Space}, that is, if given any distinct points $x,y\in X_{LC}$, there exist disjoint 
sets $U,V\in\tau$ such that, $U\cap V=\emptyset$ (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 concept to the locally compact Hausdorff space is that of a \emph{locally compact (topological)
groupoid}, which is a major concept for realizing extended quantum symmetries in
terms of \emph{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 \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 algebraic topology 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}
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