<i>F</i>^′-spaces and their product with<i>P</i>-spaces

Abstract

The F'-spaces studied here, introduced by Leonard Gillman and Melvin Henriksen, are by definition completely regular Hausdorff spaces in which disjoint cozero-sets have disjoint closures. The principal result of this paper gives a sufficient condition that a product space be an F 1 -space and shows that the condition is, in a strong sense, best possible. A fortuitous corollary in the same vein responds to a question posed by Gillman: When is a product space basically disconnected (in the sense that each of its cozero-sets has open closure)?