## You are here

Homelocally compact Hausdorff spaces

## Primary tabs

# 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 $(X_{{LC}},\tau)$ 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.

# 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.

## Mathematics Subject Classification

55U40*no label found*55-00

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections