Searching for Absolute $\mathcal{C}\mathcal{R}$ -Epic Spaces

Barr, Michael ; Kennison, John F. ; Raphael, R.

Canadian journal of mathematics, Tome 59 (2007) no. 3, pp. 465-487

Résumé

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 $\mathcal{C}\mathcal{R}$ of commutative rings (with units). We call such an embedding a $\mathcal{C}\mathcal{R}$ -epic embedding and we say that $X$ is absolute $\mathcal{C}\mathcal{R}$ -epic if every embedding of $X$ is $\mathcal{C}\mathcal{R}$ -epic. We continue this investigation. Our most notable result shows that a Lindelöf space $X$ is absolute $\mathcal{C}\mathcal{R}$ -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 Lindelöf property leads to a new class with the same closure properties that is also closed under finite products. Moreover, all $\sigma $ -compact spaces and all Lindelöf $P$ -spaces satisfy this stronger condition. We get some results in the non-Lindelöf case that are sufficient to show that the Dieudonné plank and some closely related spaces are absolute $\mathcal{C}\mathcal{R}$ -epic.

Détail

DOI : 10.4153/CJM-2007-020-9

Mots-clés : 18A20, 54C45, 54B30, absolute, ᘓR-epics, countable neighbourhood property, amply Lindelöf, Dieudonné plank

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