Connectedness modulo a topological property

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.

💡 Research Summary

The paper introduces a new notion of “𝒫‑connectedness” that generalizes ordinary connectedness by incorporating an arbitrary topological property 𝒫. A space X is defined to be 𝒫‑connected if there do not exist two disjoint cozero‑sets C and D whose closures both fail to have property 𝒫, while the remainder X \ (C ∪ D) is contained in a cozero‑set whose closure does satisfy 𝒫. When 𝒫 is the trivial property “being empty,” this definition collapses to the classical notion of connectedness, showing that the new concept truly extends the familiar one.

The authors first establish basic properties of 𝒫‑connectedness and identify mild conditions on 𝒫 that are sufficient for a clean characterization in completely regular spaces. The key hypotheses are: (i) 𝒫 is closed under taking closures, (ii) 𝒫 is preserved by cozero‑sets, and (iii) 𝒫 is hereditary for subspaces. Under these assumptions, they prove a striking theorem: a completely regular space X is 𝒫‑connected if and only if the closure in the Stone–Čech compactification βX of the set βX \ υX (where υX denotes the Hewitt realcompactification) is a connected subspace of βX. Symbolically, \