Mesocompact space
In mathematics, in the field of general topology, a topological space is said to be mesocompact if every open cover has a compact-finite open refinement.[1] That is, given any open cover, we can find an open refinement with the property that every compact set meets only finitely many members of the refinement.[2]
The following facts are true about mesocompactness:
- Every compact space, and more generally every paracompact space is mesocompact. This follows from the fact that any locally finite cover is automatically compact-finite.
- Every mesocompact space is metacompact, and hence also orthocompact. This follows from the fact that points are compact, and hence any compact-finite cover is automatically point finite.
Notes
- Hart, Nagata & Vaughan, p200
- Pearl, p23
gollark: Lyricly is now permanently banned. How fun.
gollark: <@319753218592866315> Have you read Kevin Underhill's analysis of German beekeeping laws?
gollark: > Note: right now autofree is hidden behind the -autofree flag. It will be enabled by default in V 0.3. If autofree is not used, V programs will leak memory.
gollark: You can't use the accursed resultoptional hybrid they have in situations when you have a value which *can* actually be nonexistent for whatever reasons, and it's actually `Result<T, string>` constantly.
gollark: Because they aren't really doing "option" and "result" at that point as much as a bizarre special-cased thing which is basically just indirected exceptions.
References
- K.P. Hart; J. Nagata; J.E. Vaughan, eds. (2004), Encyclopedia of General Topology, Elsevier, ISBN 0-444-50355-2
- Pearl, Elliott, ed. (2007), Open Problems in Topology II, Elsevier, ISBN 0-444-52208-5
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