Connectedness modulo an ideal

TL;DR

This paper introduces a generalized notion of connectedness in topological spaces relative to an ideal, linking it to properties of the Stone–Čech compactification and providing characterizations for specific ideals.

Contribution

It defines connectedness modulo an ideal and establishes its equivalence to connectedness of certain subspaces of the Stone–Čech compactification, extending classical concepts.

Findings

Connectedness modulo an ideal generalizes classical connectedness.

Characterizations involve the connectedness of specific subspaces of βX.

Results apply to ideals generated by open subspaces with pseudocompact closure and closed realcompact subspaces.

Abstract

For a topological space $X$ and an ideal $\mathscr{H}$ of subsets of $X$ we introduce the notion of connectedness modulo $\mathscr{H}$. This notion of connectedness naturally generalizes the notion of connectedness in its usual sense. In the case when $X$ is completely regular, we introduce a subspace $\gamma_{\mathscr H} X$ of the Stone--\v{C}ech compactification $\beta X$ of $X$, such that connectedness modulo ${\mathscr H}$ is equivalent to connectedness of $\beta X\setminus\gamma_{\mathscr H} X$. In particular, we prove that when ${\mathscr H}$ is the ideal generated by the collection of all open subspaces of $X$ with pseudocompact closure, then $X$ is connected modulo ${\mathscr H}$ if and only if $\mathrm{cl}_{\beta X}(\beta X\setminus\upsilon X)$ is connected, and when $X$ is normal and ${\mathscr H}$ is the ideal generated by the collection of all closed realcompact subspaces of $X$, then $X$ is connected modulo ${\mathscr H}$ if and only if $\mathrm{cl}_{\beta X}(\upsilon X\setminus X)$ is connected. Here $\upsilon X$ is the Hewitt realcompactification of $X$.