nLab orthocompact space

Definition

A family of subsets of a topological space, for which for every point the intersection of subsets containing it is open, is called an inner-preserving family or Q family.

A topological space, for which every (countable) open covering has an inner-preserving open refinement, is called (countably) orthocompact.

Properties

Orthocompactness generalizes other notations of compactness:

• Compact spaces are orthocompact as finite open refinements are inner-preserving in particular.
• (Countably) metacompact spaces are (countably) orthocompact as point finite open refinements are inner-preserving in particular.
• (Countably) paracompact spaces are (countably) orthocompact as locally finite open refinements are inner-preserving in particular.

Proposition

Closed subsets of (countably) orthocompact spaces are (countably) orthocompact.

Proposition

Orthocompact spaces are countably orthocompact and countably orthocompact Lindelöf spaces are orthocompact.

Proof

Follows directly from the fact that every open cover of a Lindelöf space has a countable sub-cover.

Proposition

P-spaces are countably orthocompact.

Proof

Follows directly from the fact that every countable open cover of a P-space is already inner-preserving.

(Dontchev 98)

Proposition

Spaces with Alexandroff topology are orthocompact.

Proof

Follows directly from the fact that every open cover of a space with Alexandroff topology is already inner-preserving.

(Dontchev 98)

Theorem

Given an orthocompact space $X$, the product space $X\times[0,1]$ is orthocompact iff $X$ is countably metacompact.

(Scott 75)

References

• Julian Dontchev, Orthocompactness and semi-stratifiability in the density topology. arXiv:math/9809069
• B. M. Scott, Towards a product theory for orthocompactness, “Studies in Topology”, N.M. Stavrakas and K.R. Allen, eds (1975), 517–537.