Paracompact subspaces in the box product topology
Abstract: In 1975 E. K. van Douwen showed that if is a family of Hausdorff spaces such that all finite subproducts are paracompact, then for each element of the box product the -product is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result:
Let be an infinite cardinal number, and let be a family of compact Hausdorff spaces. Let be a fixed point. Given a family of open subsets of which covers , there exists an open locally finite in refinement of which covers .We also prove a slightly weaker version of this theorem for Hausdorff spaces with ``all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen's question.
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Keywords: Paracompact space, box product
Received by editor(s): June 9, 1993
Additional Notes: The first author’s research was supported in part by NSF Grant DMS-8901931.
Communicated by: Franklin D. Tall
Article copyright: © Copyright 1996 American Mathematical Society