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

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Abstract | References | Similar Articles | Additional Information

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 . *

**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****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****K**C. J. Knight,*Box topologies*, Quart. J. Math. Oxford Ser. (2)**15**(1964), 41–54. MR**0160184****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****W**S. W. Williams,*Paracompact sets in box products*, preprint.

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Additional Information

**Peter Nyikos**

Email:
nyikos@math.sc.edu

**Leszek Piatkiewicz**

Email:
leszek@nat.pembroke.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03359-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