Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Paracompact subspaces in the box product topology

Authors: Peter Nyikos and Leszek Piatkiewicz
Journal: Proc. Amer. Math. Soc. 124 (1996), 303-314
MSC (1991): Primary 54D18; Secondary 54B10
MathSciNet review: 1327033
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In 1975 E. K. van Douwen showed that if $( X_n )_{ n \in \omega }$ is a family of Hausdorff spaces such that all finite subproducts $\prod _{ n < m } X_n$ are paracompact, then for each element $x$ of the box product $\square _{n \in \omega } X_n$ the $\sigma $-product $\sigma ( x ) = \{ y \in \square _{n \in \omega } X_n : \{ n \in \omega : x (n) \neq y (n) \} \text{ is finite} \}$ 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:

Theorem. Let $\kappa $ be an infinite cardinal number, and let $( X_{\alpha } )_{\alpha \in \kappa }$ be a family of compact Hausdorff spaces. Let $x \in \square = \square _{\alpha \in \kappa } X_\alpha $ be a fixed point. Given a family $\mathcal{R}$ of open subsets of $\square $ which covers $\sigma ( x )$, there exists an open locally finite in $\square $ refinement $\mathcal{S}$ of $\mathcal{R} $ which covers $\sigma ( x )$.

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.

References [Enhancements On Off] (What's this?)

  • vD Eric K. van Douwen, The box product of countably many metrizable spaces need not be normal, Fund. Math. 88 (1975), no. 2, 127–132. MR 0385781 (52 #6640)
  • E Ryszard Engelking, General topology, PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. MR 0500780 (58 #18316b)
  • K C. J. Knight, Box topologies, Quart. J. Math. Oxford Ser. (2) 15 (1964), 41–54. MR 0160184 (28 #3398)
  • L L. B. Lawrence, Failure of normality in the box product of uncountably many real lines, preprint. CMP 95:04.
  • vM J. van Mill, Collected papers of Eric K. van Douwen, preprint.
  • R Mary Ellen Rudin, Lectures on set theoretic topology, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Wyoming, Laramie, Wyo., August 12–16, 1974; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 23. MR 0367886 (51 #4128)
  • W S. W. Williams, Paracompact sets in box products, preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 54D18, 54B10

Retrieve articles in all journals with MSC (1991): 54D18, 54B10

Additional Information

Peter Nyikos

Leszek Piatkiewicz

PII: S 0002-9939(96)03359-X
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

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia