Products of Michael spaces and completely metrizable spaces

Authors
Citation
Dk. Burke et R. Pol, Products of Michael spaces and completely metrizable spaces, P AM MATH S, 129(5), 2001, pp. 1535-1544
Citations number
20
Categorie Soggetti
Mathematics
Journal title
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
ISSN journal
00029939 → ACNP
Volume
129
Issue
5
Year of publication
2001
Pages
1535 - 1544
Database
ISI
SICI code
0002-9939(2001)129:5<1535:POMSAC>2.0.ZU;2-B
Abstract
For disjoint subsets A, C of [0, 1] the Michael space M(A, C) = A boolean O R 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 normal ity of the product of Michael spaces with complete metric spaces. There is a ZFC example of a Lindelof Michael space M(A, C), of minimal weight N-1, w ith M(A, C) x B(N-0) Lindelof but with M(A, C) x B(N-1) not normal. (B(N-al pha) denotes the countable product of a discrete space of cardinality N-alp ha.) If M(A) denotes M(A, [0, 1] \ A), the normality of M(A) x B(N-o) impli es the normality of M(A) x S for any complete metric space S (of arbitrary weight). However, the statement "M(A, C) x B(N-1) normal implies M(A, C) x B(N-2) normal" is axiom sensitive.