Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A theorem on function spaces

Authors: Jan Baars, Joost de Groot and Jan van Mill
Journal: Proc. Amer. Math. Soc. 105 (1989), 1020-1024
MSC: Primary 54C35; Secondary 54A25
MathSciNet review: 943792
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ and $ Y$ be normal and first countable spaces, such that $ {C_p}(X)$ and $ {C_p}(Y)$ are linearly homeomorphic. Suppose $ {X^{(\alpha )}}$ is countably compact for some $ \alpha < {\omega _1}$. We prove that if $ \alpha = 1$ then $ {Y^{(\alpha )}}$ is also countably compact. The first countability condition in this result is essential. We also present examples that if $ \alpha $ is not a prime component, then $ {Y^{(\alpha )}}$ need not to be countably compact.

References [Enhancements On Off] (What's this?)

  • [1] A. V. Arhangelskii, On linear homeomorphisms of function spaces, Soviet Math. Dokl. 25 (1982), 852-855.
  • [2] Jan Baars and Joost de Groot, An isomorphical classification of function spaces of zero-dimensional locally compact separable metric spaces, Comment. Math. Univ. Carolin. 29 (1988), no. 3, 577–595. MR 972840
  • [3] J. Baars, J. de Groot, J. van Mill and J. Pelant, On topological and linear homeomorphisms of certain function spaces (to appear in top. and appl.).
  • [4] Kazimierz Kuratowski and Andrzej Mostowski, Set theory, Second, completely revised edition, North-Holland Publishing Co., Amsterdam-New York-Oxford; PWN—Polish Scientific Publishers, Warsaw, 1976. With an introduction to descriptive set theory; Translated from the 1966 Polish original; Studies in Logic and the Foundations of Mathematics, Vol. 86. MR 0485384
  • [5] Z. Semadeni, Banach spaces of continuous functions, PWN Warszawa, 1971.
  • [6] W. Sierpinski, Cardinal and ordinal numbers, PWN Warszawa, 1958.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54C35, 54A25

Retrieve articles in all journals with MSC: 54C35, 54A25

Additional Information

Keywords: Function spaces
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society