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

Barr, Michael ; Raphael, R. ; Woods, R. G.

Canadian journal of mathematics, Tome 57 (2005) no. 6, pp. 1121-1138

Résumé

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.

Détail

DOI : 10.4153/CJM-2005-043-2

Mots-clés : 18A20, 54C45, 54B30

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