A weakly infinite-dimensional space whose product with the irrationals is strongly infinite-dimensional

Pol, Elżbieta

doi:10.1090/S0002-9939-1986-0854045-0

A weakly infinite-dimensional space whose product with the irrationals is strongly infinite-dimensional
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Proc. Amer. Math. Soc. 98 (1986), 349-352 Request permission

Abstract:

We give an example of a weakly infinite-dimensional space $X$ such that the product $X \times B$ of $X$ and a subspace $B$ of the irrationals is strongly infinite-dimensional; under the assumption of the Continuum Hypothesis, $B$ can be the irrationals. This example answers a question of Addis and Gresham [AG].

References

David F. Addis and John H. Gresham, A class of infinite-dimensional spaces. I. Dimension theory and Alexandroff’s problem, Fund. Math. 101 (1978), no. 3, 195–205. MR 521122, DOI 10.4064/fm-101-3-195-205

Introduction to dimension theory

Ryszard Engelking, Teoria wymiaru, Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51], Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). MR 0482696
Ryszard Engelking and Elżbieta Pol, Countable-dimensional spaces: a survey, Dissertationes Math. (Rozprawy Mat.) 216 (1983), 41. MR 722011
William E. Haver, A covering property for metric spaces, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974, pp. 108–113. MR 0365504
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
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 152985, DOI 10.1090/S0002-9904-1963-10931-3
Roman Pol, A weakly infinite-dimensional compactum which is not countable-dimensional, Proc. Amer. Math. Soc. 82 (1981), no. 4, 634–636. MR 614892, DOI 10.1090/S0002-9939-1981-0614892-2
Roman Pol, A remark on $A$-weakly infinite-dimensional spaces, Topology Appl. 13 (1982), no. 1, 97–101. MR 637431, DOI 10.1016/0166-8641(82)90011-6
Leonard R. Rubin, R. M. Schori, and John J. Walsh, New dimension-theory techniques for constructing infinite-dimensional examples, General Topology Appl. 10 (1979), no. 1, 93–102. MR 519716, DOI 10.1016/0016-660x(79)90031-x