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

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Abstract: Let and be normal and first countable spaces, such that and are linearly homeomorphic. Suppose is countably compact for some . We prove that if then is also countably compact. The first countability condition in this result is essential. We also present examples that if is not a prime component, then need not to be countably compact.

**[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.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1989-0943792-0

Keywords:
Function spaces

Article copyright:
© Copyright 1989
American Mathematical Society