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# On $\mathcal{CR}$-epic Embeddings and Absolute $\mathcal{CR}$-epic Spaces

Published:2005-12-01
Printed: Dec 2005
• Michael Barr
• R. Raphael
• R. G. Woods
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## Abstract

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 \good. The simplest examples of \good spaces are $\sigma$-compact locally compact spaces and \Lin $P$-spaces. We show that \good first countable spaces must be locally compact. However, a bad'' class of \good spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not \good abound, and some are presented.
 MSC Classifications: 18A20 - Epimorphisms, monomorphisms, special classes of morphisms, null morphisms 54C45 - $C$- and $C^*$-embedding 54B30 - Categorical methods [See also 18B30]

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