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
gollark: It annoys me that the government goes on about how amazingly important it is and how it would be unethical to make people not go to school for a bit.
gollark: Probably people with compromised immune systems or something should avoid school.
gollark: * pretty much zero chance of dying without preexisting conditions.
gollark: I mean, on the plus side, us student-aged people aren't very affected. On the minus side, we can still transmit it...
gollark: Each [TIME UNIT] of lockdown imposes significant costs™, so clearly the solution is PWM?

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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