Products of Michael spaces and completely metrizable spaces
Authors:
Dennis K. Burke and Roman Pol
Journal:
Proc. Amer. Math. Soc. 129 (2001), 15351544
MSC (1991):
Primary 54E50, 54E52, 54D15
Published electronically:
October 10, 2000
MathSciNet review:
1712941
Fulltext PDF Free Access
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Abstract: For disjoint subsets of the Michael space has the topology obtained by isolating the points in and letting the points in retain the neighborhoods inherited from . We study normality of the product of Michael spaces with complete metric spaces. There is a ZFC example of a Lindelöf Michael space , of minimal weight , with Lindelöf but with not normal. ( denotes the countable product of a discrete space of cardinality .) If denotes , the normality of implies the normality of for any complete metric space (of arbitrary weight). However, the statement `` normal implies normal'' is axiom sensitive.
 [E]
Ryszard
Engelking, General topology, 2nd ed., Sigma Series in Pure
Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from
the Polish by the author. MR 1039321
(91c:54001)
 [F1]
William
G. Fleissner, An axiom for nonseparable Borel
theory, Trans. Amer. Math. Soc. 251 (1979), 309–328. MR 531982
(83c:03043), http://dx.doi.org/10.1090/S00029947197905319829
 [F2]
William
G. Fleissner, Separation properties in Moore spaces, Fund.
Math. 98 (1978), no. 3, 279–286. MR 0478111
(57 #17600)
 [Fr1]
D.
H. Fremlin, Consequences of Martin’s axiom, Cambridge
Tracts in Mathematics, vol. 84, Cambridge University Press, Cambridge,
1984. MR
780933 (86i:03001)
 [Fr2]
D.
H. Fremlin, Measureadditive coverings and measurable
selectors, Dissertationes Math. (Rozprawy Mat.) 260
(1987), 116. MR
928693 (89e:28012)
 [H]
W. Hurewicz, Relativ Perfekte Teile von Punktmengen ung Mengen (A), Fund. Math. 12 (1928), 78109.
 [J]
Thomas
Jech, Set theory, Academic Press [Harcourt Brace Jovanovich,
Publishers], New YorkLondon, 1978. Pure and Applied Mathematics. MR 506523
(80a:03062)
 [Je]
R.
Björn Jensen, The fine structure of the constructible
hierarchy, Ann. Math. Logic 4 (1972), 229–308;
erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR 0309729
(46 #8834)
 [K]
Alexander
S. Kechris, Classical descriptive set theory, Graduate Texts
in Mathematics, vol. 156, SpringerVerlag, New York, 1995. MR 1321597
(96e:03057)
 [Ko]
George
Koumoullis, Cantor sets in Prohorov spaces, Fund. Math.
124 (1984), no. 2, 155–161. MR 774507
(86c:54038)
 [Ku]
Kenneth
Kunen, Set theory, Studies in Logic and the Foundations of
Mathematics, vol. 102, NorthHolland Publishing Co., AmsterdamNew
York, 1980. An introduction to independence proofs. MR 597342
(82f:03001)
 [Kur]
K.
Kuratowski, Topology. Vol. I, New edition, revised and
augmented. Translated from the French by J. Jaworowski, Academic Press, New
YorkLondon; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR 0217751
(36 #840)
 [L1]
L.
Brian Lawrence, The influence of a small cardinal on
the product of a Lindelöf space and the irrationals, Proc. Amer. Math. Soc. 110 (1990), no. 2, 535–542. MR 1021211
(90m:54014), http://dx.doi.org/10.1090/S00029939199010212114
 [L2]
L.
Brian Lawrence, A ZFC example (of minimum weight) of a
Lindelöf space and a completely metrizable space with a nonnormal
product, Proc. Amer. Math. Soc.
124 (1996), no. 2,
627–632. MR 1273506
(96d:54005), http://dx.doi.org/10.1090/S000299399602864X
 [M1]
E.
Michael, The product of a normal space and a
metric space need not be normal, Bull. Amer.
Math. Soc. 69
(1963), 375–376. MR 0152985
(27 #2956), http://dx.doi.org/10.1090/S000299041963109313
 [M2]
Ernest
A. Michael, Paracompactness and the Lindelöf property in
finite and countable Cartesian products, Compositio Math.
23 (1971), 199–214. MR 0287502
(44 #4706)
 [M3]
E.
Michael, Correction to: “A note on
completely metrizable spaces” [Proc.\ Amer.\ Math.\ Soc.\ {96}
(1986), no.\ 3, 513–522; MR0822451 (87f:54045)], Proc. Amer. Math. Soc. 100 (1987), no. 1, 204. MR 883430
(88a:54068), http://dx.doi.org/10.1090/S00029939198708834307
 [P]
Roman
Pol, Note on decompositions of metrizable spaces. II, Fund.
Math. 100 (1978), no. 2, 129–143. MR 0494011
(58 #12953)
 [RS]
Mary
Ellen Rudin and Michael
Starbird, Products with a metric factor, General Topology and
Appl. 5 (1975), no. 3, 235–248. MR 0380709
(52 #1606)
 [S]
A.
H. Stone, On 𝜎discreteness and Borel isomorphism,
Amer. J. Math. 85 (1963), 655–666. MR 0156789
(28 #33)
 [E]
 R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989. MR 91c:54001
 [F1]
 W. Fleissner, An axiom for nonseparable Borel theory, Trans. Amer. Math. Soc. 251 (1979), 309328. MR 83c:03043
 [F2]
 W. Fleissner, Separation properties in Moore spaces, Fund. Math. 98 (1978), 279286. MR 57:17600
 [Fr1]
 D. H. Fremlin, Consequences of Martin's Axiom, Cambridge University Press, Cambridge, 1984. MR 86i:03001
 [Fr2]
 D. H. Fremlin, Measureadditive coverings and measurable selectors, Dissertationes Math. 260 (1987). MR 89e:28012
 [H]
 W. Hurewicz, Relativ Perfekte Teile von Punktmengen ung Mengen (A), Fund. Math. 12 (1928), 78109.
 [J]
 T. Jech, Set Theory, Academic Press, New York, 1978. MR 80a:03062
 [Je]
 R. B. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229308. MR 46:8834
 [K]
 A. S. Kechris, Classical Descriptive Set Theory, SpringerVerlag, New York, 1995. MR 96e:03057
 [Ko]
 G. Koumoullis, Cantor sets in Prohorov spaces, Fund. Math. 124 (1984), 155161. MR 86c:54038
 [Ku]
 K. Kunen, Set Theory, NorthHolland, Amsterdam, 1980. MR 82f:03001
 [Kur]
 K. Kuratowski, Topology, Volume 1, Polish Scientific Publishers, Warsaw, 1966. MR 36:840
 [L1]
 L. B. Lawrence, The influence of a small cardinal on the product of a Lindelöf space with the irrationals, Proc. Amer. Math. Soc. 110 (1990), 535542. MR 90m:54014
 [L2]
 L. B. Lawrence, A ZFC example (of minimum weight) of a Lindelöf space and a completely metrizable space with a nonnormal product, Proc. Amer. Math. Soc. 124 (1996), 627632. MR 96d:54005
 [M1]
 E. A. Michael, The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc. 69 (1963), 375376. MR 27:2956
 [M2]
 E. A. Michael, Paracompactness and the Lindelöf property in finite and countable cartesian products, Compositio Math. 23 (1971), 199214. MR 44:4706
 [M3]
 E. A. Michael, A note on completely metrizable spaces, Proc. Amer. Math. Soc. 96 (1986), 513522. MR 88a:54068
 [P]
 R. Pol, Note on decompositions of metrizable spaces II, Fund. Math. 100 (1978), 129143. MR 58:12953
 [RS]
 M. E. Rudin and M. Starbird, Products with a metric factor, Gen. Top. Appl. 5 (1975), 235248. MR 52:1606
 [S]
 A. Stone, On discreteness and Borel isomorphisms, Amer. J. Math. 85 (1963), 655666. MR 28:33
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Additional Information
Dennis K. Burke
Affiliation:
Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056
Email:
dburke@miavx1.muohio.edu
Roman Pol
Affiliation:
Department of Mathematics, Warsaw University, Warsaw, Poland
Email:
pol@mimuw.edu.pl
DOI:
http://dx.doi.org/10.1090/S0002993900056641
PII:
S 00029939(00)056641
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
