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Paracompactness in box products. (English) Zbl 0568.54011
The author presents several results on paracompactness in box products with uncountably many factors. He uses the following notation. Suppose \(\kappa\) is a cardinal and for each \(\alpha\in \kappa\), \(X_{\alpha}\) is a space. Any subset of \(\Pi_{\alpha \in \kappa}X_{\alpha}\) of the form \(B=\Pi_{\alpha \in \kappa}B_{\alpha}\) where each \(B_{\alpha}\) is open in \(X_{\alpha}\) is called an open box. The support of B is defined as \(spt(B)=\{\alpha \in \kappa: B_{\alpha}\neq X_{\alpha}\}.\) If \(\lambda\) is a cardinal, then \(<\lambda -\square_{\alpha \in \kappa}X_{\alpha}\) denotes \(\Pi_{\alpha \in \kappa}X_{\alpha}\) with the topology having as base the set of all open boxes B with \(spt(B)\in [\kappa]<\lambda.\) If \(\kappa <\lambda\), then \(\square_{\alpha \in \kappa}X_{\alpha}\) is used to denote \(<\lambda -\square_{\alpha \in \kappa}X_{\alpha}.\) Recall that a space is called a P-space if each \(G_{\delta}\) subset is open. The author’s first main result is obtained in ZFC. Theorem. If \(X_{\alpha}\) is a paracompact P-space for each \(\alpha \in \omega_ 1\), then \(<\omega_ 1-\square_{\alpha \in \kappa}X_{\alpha}\) is paracompact. This may be compared to the result of E. K. van Douwen for \(\omega_{\mu}\)-metrizable spaces [see General Topol. Appl. 7, 71-76 (1977; Zbl 0341.54008)]. His second main result is this. Theorem (CH). If \(X_{\alpha}\) is a paracompact locally compact scattered space for each \(\alpha \in \omega_ 1\), then \(<\omega_ 1-\square_{\alpha \in \kappa}X_{\alpha}\) is paracompact. The author constructs an example of a compact orderable space T such that \(\square^{\omega}T\) is not paracompact. The construction differs from that used in obtaining previous examples of nonparacompact box products (see the author’s survey article ”Box Products” in Handbook of set- theoretic topology, 169-201 (1984; Zbl 0565.54007)]. The author concludes with a discussion of the problem of the paracompactness of the full box product of uncountably many compact first countable spaces.
Reviewer: H.H.Wicke
54B10 Product spaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54G20 Counterexamples in general topology
54A35 Consistency and independence results in general topology