Metacompactness: every open cover has a point-finite open refinement

A topological space $X$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison) is metacompact when every open cover of $X$ has a point-finite open refinement: for every open cover $\mathcal{U}$ of $X$ there is a family $\mathcal{V}$ of open sets such that $\mathcal{V}$ covers $X$, every $V\in \mathcal{V}$ is contained in some $U\in \mathcal{U}$, and every point of $X$ belongs to only finitely many members of $\mathcal{V}$ (Refinements, locally finite families, point-finite families, and star refinements).

Point-finiteness is the only new component. Refinement and covering are those of Refinements, locally finite families, point-finite families, and star refinements; metacompactness weakens paracompactness (Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word) by asking the refining family to be point-finite rather than locally finite. Local finiteness implies point finiteness, so every paracompact space is metacompact, and no separation axiom is built into the word.

• Why this item exists on this page. The ZF countermodel of Stone's theorem on this page produces a metrizable space with an open cover that has no point-finite open refining cover; that is the precise failure, and it is what the relative-consistency theorem and the open-status remark state. (The empty family is a point-finite refinement in the bare containment sense, but it does not cover a nonempty space.) The word metacompact is used only as an abbreviation for that covering property.

• Effectivity is a separate strengthening. A refinement is called effective when it comes equipped with a refinement map $a:\mathcal{V}\to \mathcal{U}$ satisfying $V\subseteq a(V)$; the strengthening that every open cover of every discrete metric space has an effective point-finite open refinement is equivalent to the Axiom of Choice and is treated as its own theorem below, not as part of this definition.