# Searching for more absolute CR-epic spaces

## Michael Barr, John F. Kennison, and R. Raphael

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 Dieudonne construction, and yields absolute CR-epic spaces which are not Lindelof. For the Lindelof 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 $G_\delta$ sets has a countable subcover, provided each compact subset can be covered by a finite subset.) In this paper, we find further properties of Lindelof 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.

Keywords: absolute CR-epics, countable neighbourhood property, Alster's condition Dieudonne plank

2000 MSC: 18A20, 54C45, 54B30

Theory and Applications of Categories, Vol. 22, 2009, No. 3, pp 54-76.

http://www.tac.mta.ca/tac/volumes/22/3/22-03.dvi
http://www.tac.mta.ca/tac/volumes/22/3/22-03.ps
http://www.tac.mta.ca/tac/volumes/22/3/22-03.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/3/22-03.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/3/22-03.ps

Revised 2009-07-15. Original version at
http://www.tac.mta.ca/tac/volumes/22/3/22-03a.dvi

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