Searching for absolute CR-epic spaces

In previous papers, [Barr, Burgess, & Raphael (2003), Barr, Raphael, & Woods (to appear)], two of us have investigated the situation of a topo-logical space Y and a subspace X such that the induced map C(Y) // 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 Lindelöf space X is absolute CR-epic if a countable intersection of βX-neighbourhoods of X is a βX-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 σ-compact spaces and all Lin-delöf P-spaces satisfy this stronger condition. We get some results in the non-Lindelöf case that are sufficient to show that the Dieudonne ́ plank and some closely related spaces are absolute CR-epic