The separated cellularity of a topological space and finite separation spaces

Abstract

A separated cell A of a topological space X is a family of pairwise disjoint open sets with the property that, for any partition A= A 1∪ A 2 of A , ∪ A 1 and ∪ A 2 are completely separated. The separated cellularity sc( X) of X is the supremum of all cardinals of separated cells. A space X has finite separation if it has no infinite separated cells. With X compact, every separated cell of X has size < κ if and only if no regular closed subset has a continuous surjection onto βκ. 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 dyadic spaces have finite separation. Likewise, every compact scattered space, every compact countably tight space, and every compact hereditarily paracompact space has finite separation.