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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
##### MSC:
 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