Real Algebraic and Analytic Geometry
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Submission: 2004, October 6.
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. First countable spaces must be locally compact in order to be \good. 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. Many open questions remain.
Mathematics Subject Classification (2000): 18A20, 54C45, 54B30.
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