Searching for More Absolute Cr-epic Spaces

We continue our examination of absolute CR -epic spaces, or spaces with the property that any embedding induces an epimorphism, in the category of commutative rings, between their rings of continuous functions. We examine more closely the deleted plank construction, which generalizes the Dieudonné construction, and yields absolute CR -epic spaces which are not Lindelöf. For the Lindelöf case, an earlier paper has shown the usefulness of the countable neighbourhood property, CNP, and the Alster condition (where CNP means that the space is a P-space in any compactification and the Alster condition says that any cover of the space by Gδ sets has a countable subcover, provided each compact subset can be covered by a finite subset.) In this paper, we find further properties of Lindelöf CNP spaces and of Alster spaces, including constructions that preserve these properties and conditions equivalent to these properties. We explore the outgrowths of such spaces and find several examples that answer open questions in our previous work.