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Products of Michael spaces and completely metrizable spaces

Authors: Dennis K. Burke and Roman Pol
Journal: Proc. Amer. Math. Soc. 129 (2001), 1535-1544
MSC (1991): Primary 54E50, 54E52, 54D15
Published electronically: October 10, 2000
MathSciNet review: 1712941
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Abstract: For disjoint subsets $A,C$ of $[0,1]$ the Michael space $M(A,C)=A\cup C$ has the topology obtained by isolating the points in $C$ and letting the points in $A$ retain the neighborhoods inherited from $[0,1]$. We study normality of the product of Michael spaces with complete metric spaces. There is a ZFC example of a Lindelöf Michael space $M(A,C)$, of minimal weight $\aleph_1$, with $M(A,C)\times B(\aleph_0)$ Lindelöf but with $M(A,C)\times B(\aleph_1)$ not normal. ( $B(\aleph_\alpha)$ denotes the countable product of a discrete space of cardinality $\aleph_\alpha$.) If $M(A)$ denotes $M(A,[0,1]\smallsetminus A)$, the normality of $M(A)\times B(\aleph_0)$ implies the normality of $M(A)\times S$ for any complete metric space $S$ (of arbitrary weight). However, the statement `` $M(A,C)\times B(\aleph_1)$ normal implies $M(A,C)\times B(\aleph_2)$ normal'' is axiom sensitive.

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Additional Information

Dennis K. Burke
Affiliation: Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056

Roman Pol
Affiliation: Department of Mathematics, Warsaw University, Warsaw, Poland

Keywords: Michael space, product spaces, normal, completely metrizable, Baire space, absolute $G_\delta$
Received by editor(s): March 8, 1998
Received by editor(s) in revised form: July 28, 1999
Published electronically: October 10, 2000
Additional Notes: The results in this note were obtained while the second author was a Visiting Professor at Miami University. The author would like to express his gratitude to the Department of Mathematics and Statistics of Miami University for their hospitality.
Communicated by: Alan Dow
Article copyright: © Copyright 2000 American Mathematical Society