A weakly infinitedimensional space whose product with the irrationals is strongly infinitedimensional
Author:
Elżbieta Pol
Journal:
Proc. Amer. Math. Soc. 98 (1986), 349352
MSC:
Primary 54F45; Secondary 54B10
MathSciNet review:
854045
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Abstract: We give an example of a weakly infinitedimensional space such that the product of and a subspace of the irrationals is strongly infinitedimensional; under the assumption of the Continuum Hypothesis, can be the irrationals. This example answers a question of Addis and Gresham [AG].
 [AG]
David
F. Addis and John
H. Gresham, A class of infinitedimensional spaces. I. Dimension
theory and Alexandroff’s problem, Fund. Math.
101 (1978), no. 3, 195–205. MR 521122
(80b:54041)
 [AP]
P. S. Alexandroff and B. A. Pasynkov, Introduction to dimension theory, Nauka, Moskva, 1973. (Russian)
 [E]
Ryszard
Engelking, Dimension theory, NorthHolland Publishing Co.,
Amsterdam, 1978. Translated from the Polish and revised by the author;
NorthHolland Mathematical Library, 19. MR 0482697
(58 #2753b)
 [EP]
Ryszard
Engelking and Elżbieta
Pol, Countabledimensional spaces: a survey, Dissertationes
Math. (Rozprawy Mat.) 216 (1983), 41. MR 722011
(85f:54075)
 [H]
William
E. Haver, A covering property for metric spaces, Topology
Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va.,
1973), Springer, Berlin, 1974, pp. 108–113. Lecture Notes in
Math., Vol. 375. MR 0365504
(51 #1756)
 [K]
K.
Kuratowski, Topology. Vol. I, New edition, revised and
augmented. Translated from the French by J. Jaworowski, Academic Press, New
York, 1966. MR
0217751 (36 #840)
 [M]
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
 [P1]
Roman
Pol, A weakly infinitedimensional
compactum which is not countabledimensional, Proc. Amer. Math. Soc. 82 (1981), no. 4, 634–636. MR 614892
(82f:54059), http://dx.doi.org/10.1090/S00029939198106148922
 [P2]
Roman
Pol, A remark on 𝐴weakly infinitedimensional spaces,
Topology Appl. 13 (1982), no. 1, 97–101. MR 637431
(83b:54051), http://dx.doi.org/10.1016/01668641(82)900116
 [RSW]
Leonard
R. Rubin, R.
M. Schori, and John
J. Walsh, New dimensiontheory techniques for constructing
infinitedimensional examples, General Topology Appl.
10 (1979), no. 1, 93–102. MR 519716
(80e:54049)
 [AG]
 D. F. Addis and J. H. Gresham, A class of infinitedimensional spaces. Part I: Dimension theory and Alexandroff 's problem, Fund. Math. 101 (1978), 195205. MR 521122 (80b:54041)
 [AP]
 P. S. Alexandroff and B. A. Pasynkov, Introduction to dimension theory, Nauka, Moskva, 1973. (Russian)
 [E]
 R. Engelking, Dimension theory, PWN, Warszawa, 1978. MR 0482697 (58:2753b)
 [EP]
 R. Engelking and E. Pol, Countabledimensional spaces: A survey, Dissertationes Math. 216 (1983), 141. MR 722011 (85f:54075)
 [H]
 W. E. Haver, A covering property for metric spaces, Lecture Notes in Math., vol. 375, SpringerVerlag, Berlin and New York, 1974, pp. 108113. MR 0365504 (51:1756)
 [K]
 K. Kuratowski, Topology, vols. I, II, PWN, Warszawa, 1966, 1968. MR 0217751 (36:840)
 [M]
 E. Michael, The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc. 69 (1963), 375376. MR 0152985 (27:2956)
 [P1]
 R. Pol, A weakly infinitedimensional compactum which is not countabledimensional, Proc. Amer. Math. Soc. 82 (1981), 634636. MR 614892 (82f:54059)
 [P2]
 , A remark on weakly infinitedimensionalspaces, Topology Appl. 13 (1982), 97101. MR 637431 (83b:54051)
 [RSW]
 L. R. Rubin, R. M. Shori, and J. J. Walsh, New dimensiontheory techniques for constructing infinitedimensional examples, General Topology Appl. 10 (1979), 93103. MR 519716 (80e:54049)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198608540450
PII:
S 00029939(1986)08540450
Keywords:
Weakly infinitedimensional spaces,
products,
property
Article copyright:
© Copyright 1986 American Mathematical Society
