Finite Separation Spaces
by
Jorge Martinez
University of Florida
Coauthors: Robert T. Finn, Warren W. McGovern, Vladimir Uspenskii

A (Tychonoff) space X has finite separation if for each infinite family U₁, U₂, ... of pairwise disjoint nonempty open sets in X there is a partition of the natural numbers into subsets A and B such that U_A = \cup _{n in A} U_n and U_B = \cup _{n in B} U_n are not completely separated. X has finite separation if and only if no regular closed subset has a retraction onto \beta\omega. It is shown that for normal, first-countable spaces finite separation is equivalent to sequential compactness. Compact finite separation spaces are examined, and compared to other classes of compact spaces which occur in the literature. It is shown that all strong \pi\chi -spaces, and therefore all dyadic spaces have finite separation. Likewise, every compact scattered space, every compact countably tight space, and every compact hereditarily paracompact space has finite separation.

Date received: May 31, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caag-04.