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 clβX(βX\υX) is connected, and that P-connectedness is preserved under perfect open continuous surjections. We leave some problems open.