Contents

# Contents

## 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.

###### 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.

###### 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.