nLab stably compact space

Idea

Stably compact spaces are topological spaces which share many of the desirable properties of compact Hausdorff spaces, such as compactness and local compactness, without being Hausdorff or even T1.

They are also a convenient setting for convergence in an ordered setting, being deeply linked to compact ordered spaces.

Definition

A topological space $X$ is called stably compact if the following conditions are met:

• $X$ is T0;
• $X$ is compact;
• $X$ is locally compact;
• $X$ is sober;
• $X$ is coherent, meaning that the intersection of two compact saturated subsets is compact.

Note that the latter notion of coherence is slightly different than the one given at coherent space.

Connection with compact ordered spaces

(…)

Connection with compact regular bitopological spaces

(…) For details see (Jung-Moshier 2006).

Related concepts

• compact ordered space
• compact Hausdorff space, compact space, locally compact space

References

• Achim Jung, Stably compact spaces and the probabilistic powerspace construction, ENTCS 87, 2004 (doi:10.1016/j.entcs.2004.10.001).

• Achim Jung and M. Andrew Moshier. On the bitopological nature of Stone duality. Technical Report CSR-06-13. School of Computer Science, University of Birmingham, December 2006, 110 pages. $[$pdf$]$