Celebratio Mathematica — Rudin

A Tribute to Mary Ellen Rudin

Works connected to Amer Bešlagić

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A. Bešlagić and M. E. Rudin: “Set-theoretic constructions of nonshrinking open covers,” Topology Appl. 20 : 2 (August 1985), pp. 167–177. MR 800847 Zbl 0574.54020 article

A family \[ \{M_{\alpha}\mid \alpha\in A\} \] is a shrinking of a cover \[ \{O_{\alpha}\mid \alpha\in A\} \] of a topological space if \[ \{M_{\alpha}\mid \alpha\in A\} \] also covers and \( M_{\alpha}\subset O_{\alpha} \) for all \( \alpha\in A \).

\( \lozenge^{++} \) implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking.

\( \lozenge \) implies that there is a \( P \)-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in [Chiba et al. 1986] to show that any space which has a normal product with every \( P \)-space is metrizable.