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Not Every K1-Embedded Subspace is K0-Embedded

Published online by Cambridge University Press:  20 November 2018

Jan van Mill
Jan van Mill*
Affiliation:
University of Wisconsin, Madison, Wisconsin
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All topological spaces under discussion are assumed to be Tychonoff.

For any topological space X let τ(X) denote the topology of X. If X ᑕ Y then a function κ : τ(X) ⟶ τ(Y) is called an extender provided that κ(U) ∩ X = U for all U ∊ τ(X). In addition, X is said to be Kn-embedded in Y (cf. [3]) provided there is an extender κ : τ(X) ⟶ τ(Y) such that

The extender κ is called a Kn-function (cf. [3]).

Eric van Douwen has asked whether there is a space X with a subspace Z which is Ki-embedded but not K0-embedded. The aim of this note is to answer this question.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 4 , 01 August 1979 , pp. 818 - 823
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Bell, M. G. and van Mill, J., The compactness number of a compact topological space (to appear in Fund. Math.).Google Scholar
2. van Douwen, E. K., Simultaneous extension of continuous functions, Thesis, Vrije Universiteit, Amsterdam (1975).Google Scholar
3. van Douwen, E. K., Simultaneous linear extension of continuous functions, Gen. Top. Appl. 5 (1975), 297–319.Google Scholar
4. van Douwen, E. K. and van Mill, J., Supercompact spaces (to appear in Gen. Top. Appl.).Google Scholar
5. de Groot, J. and Aarts, J. M., Complete regularity as a separation axiom, Can. J. Math. 21 (1969), 96–105.Google Scholar
6. Rudin, M. E., Lectures on set theoretic topology, Regional Conf. Ser. in Math. No. 23, Am. Math. Soc. (Providence, RI, 1975).Google Scholar