A note on discrete C-embedded subspaces

Abstract

It is shown that in some non-discrete topological spaces, discrete subspaces with certain cardinality are C-embedded. In particular, this generalizes the well-known fact that every countable subset of P-spaces are C-embedded. In the presence of the measurable cardinals, we observe that if X is a discrete space then every subspace of υ X (i.e., the Hewitt realcompactification of X) whose cardinal is nonmeasurable, is a C-embedded, discrete realcompact subspace of υ X. This generalizes the well-known fact that the discrete spaces with nonmeasurable cardinal are realcompact.

When is the super socle of C(X) prime?

“…It is folklore that one of the main objectives of working in the context of C(X) is to characterize topological properties of a given space X in terms of a suitable algebraic properties of C(X). It turns out, in this regard, the ideal C F (X), plays an appropriate role in the literature, see for examples [2], [3], [5], [6], [8], [9] [14], [15], and [17]. Motivated by this role of C F (X), the concept of the super socle, which contains C F (X), is introduced in [11], see also [16].…”