Connectedness modulo a topological property

Let P be a topological property. We say that a space X is P -connected if there exists no pair C and D of disjoint cozero-sets of X with non- P closure such that the remainder X \ ( C ∪ D ) is contained in a cozero-set of X with P closure. If P is taken to be “being empty” then P -connectedness coincides with connectedness in its usual sense. We characterize completely regular P -connected spaces, with P subject to some mild requirements. Then, we study conditions under which unions of P -connected subspaces of a space are P -connected. Also, we study classes of mappings which preserve P -connectedness. We conclude with a detailed study of the special case in which P is pseudocompactness. In particular, when P is pseudocompactness, we prove that a completely regular space X is P -connected if and only if cl β X ( β X \ υ X ) is connected, and that P -connectedness is preserved under perfect open continuous surjections. We leave some problems open.