Connectedness modulo a topological property

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