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# Searching for Absolute $\mathcal{CR}$-Epic Spaces

Published:2007-06-01
Printed: Jun 2007
• Michael Barr
• John F. Kennison
• R. Raphael
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## Abstract

In previous papers, Barr and Raphael investigated the situation of a topological space $Y$ and a subspace $X$ such that the induced map $C(Y)\to C(X)$ is an epimorphism in the category $\CR$ of commutative rings (with units). We call such an embedding a $\CR$-epic embedding and we say that $X$ is absolute $\CR$-epic if every embedding of $X$ is $\CR$-epic. We continue this investigation. Our most notable result shows that a Lindel\"of space $X$ is absolute $\CR$-epic if a countable intersection of $\beta X$-neighbourhoods of $X$ is a $\beta X$-neighbourhood of $X$. This condition is stable under countable sums, the formation of closed subspaces, cozero-subspaces, and being the domain or codomain of a perfect map. A strengthening of the Lindel\"of property leads to a new class with the same closure properties that is also closed under finite products. Moreover, all \s-compact spaces and all Lindel\"of $P$-spaces satisfy this stronger condition. We get some results in the non-Lindel\"of case that are sufficient to show that the Dieudonn\'e plank and some closely related spaces are absolute $\CR$-epic.
 Keywords: absolute $\mathcal{CR}$-epics, countable neighbourhoo9d property, amply Lindelöf, Diuedonné plank
 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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