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

http://dx.doi.org/10.4153/CJM-2005-043-2
Canad. J. Math. 57(2005), 1121-1138

  Published:2005-12-01
 Printed: Dec 2005

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.