Connectedness modulo an ideal

For a topological space X   and an ideal HH of subsets of X   we introduce the notion of connectedness modulo HH. 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 γHXγHX of the Stone–Čech compactification βX of X  , such that connectedness modulo HH is equivalent to connectedness of βX∖γHXβX∖γHX. In particular, we prove that when HH is the ideal generated by the collection of all open subspaces of X with pseudocompact closure, then X   is connected modulo HH if and only if clβX(βX∖υX)clβX(βX∖υX) is connected, and when X   is normal and HH is the ideal generated by the collection of all closed realcompact subspaces of X, then X   is connected modulo HH if and only if clβX(υX∖X)clβX(υX∖X) is connected. Here υX is the Hewitt realcompactification of X.