Search: MSC category 54C45 ( $C$- and $C^$-embedding *$-embedding * )  Expand all Collapse all Results 1 - 2 of 2 1. CJM 2007 (vol 59 pp. 465) Barr, Michael; Kennison, John F.; Raphael, R.  Searching for Absolute$\mathcal{CR}$-Epic Spaces In previous papers, Barr and Raphael investigated the situation of a topological space$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}$-epics, countable neighbourhoo9d property, amply LindelÃ¶f, DiuedonnÃ© plankCategories:18A20, 54C45, 54B30 2. CJM 2005 (vol 57 pp. 1121) Barr, Michael; Raphael, R.; Woods, R. G.  On$\mathcal{CR}$-epic Embeddings and Absolute$\mathcal{CR}$-epic Spaces We study Tychonoff spaces$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. Categories:18A20, 54C45, 54B30