On a Class of Countably Paracompact Spaces

In this note we shall characterize a topological property which is stronger than countable paracompactness but which is equivalent to it for normal spaces. A real valued function on a topological space X is locally bounded if each point has a neighborhood on which the function is bounded. Let C(X) denote the set of real valued continuous functions on X. A topological space is a cb-space if for each locally bounded function h, there exists /GC(X) such that/e|&|. J. G. Home, Jr. initiated a study of co-spaces and reported on his work in [2]. For /G C(X), the set on which / vanishes is called the zero-set of / and it is denoted by Z(f). The complement of a zero-set is called a cozero-set. A cozero cover is a cover consisting of cozero-sets. A family of continuous functions is locally finite if the collection of cozero-sets associated with the family is a locally finite collection of sets. A family £F of continuous functions is a partition of unity if 0 £|/ for all/G? and ]>^/eg:/(x) = 1 for all xEX. A partition of unity is subordinate to a cover if the collection of cozero-sets associated with the partition is a refinement of the cover. A countable cover { Î/B} is an increasing cover if Un C U»+x for all «. In this paper the term cover will be used to mean open cover.