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: 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.
- 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
Engelking, Topologia ogólna, Państwowe
Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna. Tom
47. [Mathematics Library. Vol. 47]. MR
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, https://doi.org/10.1093/qmath/15.1.41
- 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.
- E. K. van Douwen, The box product of countably many metrizable spaces need not be normal, Fund. Math. 88 (1975), 127--132, MR 52:6640.
- R. Engelking, General topology, PWN, Warszawa, 1977, MR 58:18316b.
- C. J. Knight, Box topologies, Quart. J. Math. Oxford 15 (1964), 41--54, MR 28:3398.
- L. B. Lawrence, Failure of normality in the box product of uncountably many real lines, preprint. CMP 95:04.
- J. van Mill, Collected papers of Eric K. van Douwen, preprint.
- M. E. Rudin, Lectures on set-theoretic topology, CBMS Regional Conf. Ser. in Math., vol. 23, Amer. Math. Soc., Providence, RI, 1975, MR 51:4128.
- S. W. Williams, Paracompact sets in box products, preprint.
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