# 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

1. ^ Hart, Nagata & Vaughan, p200
2. ^ Pearl, p23

## 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