Connectedness modulo an ideal

For a topological space $X$ and an ideal $\mathcal{H}$ of subsets of $X$ we introduce the notion of connectedness modulo $\mathcal{H}$. 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 ${\gamma }_{\mathcal{H}}X$ of the Stone–\v{C}ech compactification $\beta X$ of $X$, such that connectedness modulo $\mathcal{H}$ is equivalent to connectedness of $\beta X\setminus {\gamma }_{\mathcal{H}}X$. In particular, we prove that when $\mathcal{H}$ is the ideal generated by the collection of all open subspaces of $X$ with pseudocompact closure, then $X$ is connected modulo $\mathcal{H}$ if and only if $$\mathrm{c}\mathrm{l}${\beta X}(\beta X\setminus\upsilon X)$isconnected,andwhen$X$isnormaland${\mathscr H}$ is the ideal generated by the collection of all closed realcompact subspaces of $X$, then $X$ is connected modulo $\mathcal{H}$ if and only if $$\mathrm{c}\mathrm{l}${\beta X}(\upsilon X\setminus X)$isconnected\mathrm{.}Here$\upsilon X$ is the Hewitt realcompactification of $X$.