Metacompact space

Definition

Let $X$ be a topological space. A family $\mathcal{V}$ of subsets of $X$ is point-finite if each $x \in X$ belongs to only finitely many members of $\mathcal{V}$ . The space $X$ is metacompact if every open cover $\mathcal{U}$ of $X$ has a point-finite open refinement. That is, there is a point-finite open cover $\mathcal{V}$ such that each $V \in \mathcal{V}$ lies in some $U \in \mathcal{U}$ .

Remarks

Example: every paracompact space is metacompact. A locally finite family is point-finite, since a neighbourhood of $x$ meeting finitely many members also shows that $x$ lies in finitely many.

Non-example: the space $[0, \omega_1)$ of countable ordinals with the order topology is not metacompact. It is countably compact but not compact. A countably compact metacompact space is always compact.

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Open cover concept
Topological space concept
Metacompact space definition

Tag	`J8QBXGP7KK`
URI	`m2://J8QBXGP7KK`
Kind	definition
Writing	W0 stub
MSC	54D20
Defines	Metacompact space
Uses	Topological space, Open cover
Changed	2 November 2026

Metacompact space

Definition

Remarks

Reading path

Further reading

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