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Paracompact subspaces in the box product topology |
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| Authors: | Peter Nyikos Leszek Piatkiewicz |
| Affiliation: | Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208 ; Department of Mathematics and Computer Science, Pembroke State University, Pembroke, North Carolina 28372 |
| 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. |
| Keywords: | Paracompact space box product |
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