A characterization of covering dimension for collectionwise normal spaces

James Austin French
1970 Proceedings of the American Mathematical Society  
Ostrand [3] gave the following characterization of the covering dimension for metric spaces. A metric space X is of dimension ^w if and only if for each open cover CoiX and each integer k^n + 1 there exist k discrete families of open sets U\, U2, • • • ,Uk such that the union of any n + 1 of the Ui is a cover of X which refines C. The purpose of this paper is to generalize Ostrand's theorem to collectionwise normal spaces by using an argument similar to that of Michael [l, Theorem 2].
more » ... eorem 2]. Definitions. The collection H of subsets of the topological space X strongly refines the collection G of subsets of X if and only if for each hEH there is an element gEG such that hEg, where h denotes the closure of h.
doi:10.1090/s0002-9939-1970-0261533-5 fatcat:bsxcjvonsjhfbopkyymb3azlya