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On $\mathcal{C}\mathcal{R}$-Epic Embeddings and Absolute $\mathcal{C}\mathcal{R}$-Epic Spaces

Published online by Cambridge University Press:  20 November 2018

Michael Barr ,
R. Raphael  and
R. G. Woods
Michael Barr
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6, e-mail: [email protected]
R. Raphael
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montreal, QC, H4B 1R6, e-mail: [email protected]
R. G. Woods
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, [email protected]
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Abstract

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We study Tychonoff spaces $X$ with the property that, for all topological embeddings $X\,\to \,Y$, the induced map $C(Y)\,\to \,C(X)$ is an epimorphism of rings. Such spaces are called absolute $\mathcal{C}\mathcal{R}$-epic. The simplest examples of absolute $\mathcal{C}\mathcal{R}$-epic spaces are $\sigma $-compact locally compact spaces and Lindelöf $P$-spaces. We show that absolute $\mathcal{C}\mathcal{R}$-epic first countable spaces must be locally compact.

However, a “bad” class of absolute $\mathcal{C}\mathcal{R}$-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute $\mathcal{C}\mathcal{R}$-epic abound, and some are presented.

Keywords

18A2054C4554B30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 6 , 01 December 2005 , pp. 1121 - 1138
Copyright
Copyright © Canadian Mathematical Society 2005

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