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Compact monotonically metacompact spaces are metrizable. (English) Zbl 1264.54039

A space is countably metacompact iff every countable open cover has a point-finite open covering refinement.
Now suppose \(X\) is countably metacompact, and if \(\mathcal U\) is a countable open cover then \(r(\mathcal U)\) denotes a point-finite open covering refinement of \(\mathcal U\). Can we assign \(r\) so that if \(\mathcal V\) is a countable open cover refining the countable open cover \(\mathcal U\) then \(r(\mathcal V)\) refines \(r(\mathcal U)\)? If so, we say that \(X\) is monotonically countably metacompact.
The main theorem in this paper is that a compact Hausdorff monotonically countably metacompact space is metrizable. The proof uses the notion of caliber: \(X\) has caliber \(\kappa\) iff given an open family of size \(\geq \kappa\) there is a subcollection of size at most \(\kappa\) whose intersection is nonempty. The theorem follows from two lemmas: a compact Hausdorff monotonically countably metacompact space has caliber \(\omega_1\), and a compact Hausdorff monotonoically countably metacompact space with caliber \(\omega_1\) is metrizable.
The authors also show that neither the sequential fan nor the single ultrafilter space is monotonically countably metacompact. The sequential fan is the set \(\omega^2 \cup \{\infty\}\) where \(\infty\) is the only non-isolated point, and a neigborhood of \(\infty\) includes all but finitely many elements from each \(\{n\} \times \omega\). The single ultrafilter space is the set \(\omega \cup \{\mathcal F\}\) where \(\mathcal F\) is a non-principal ultrafilter on \(\omega\), \(\mathcal F\) is the only non-isolated point, and if \(U\) is a neighborhood of \(\mathcal F\) then there is some \(F \in \mathcal F\) with \(F \subset U\). Both of these spaces are countable and monotonically Lindelöf (similar to the definition above, but \(\mathcal U\) is an arbitrary open cover and \(r(\mathcal U)\) is a countable covering refinement of \(\mathcal U)\), thus showing that the two monotonic properties are distinct even in the class of countable Hausdorff spaces.
An open question is whether every countable monotonically countably metacompact space is metrizable.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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