ABSTRACT. We extend the well-known and important fact that "a topological space X is compact if and only if every ideal in C(X) is fixed", to more general topological spaces. Some interesting consequences are also observed. In particular, the maximality of compact Hausdorff spaces with respect to the property of compactness is generalized and the topological spaces with this generalized property are characterized.
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.
λ -Perfect maps, a generalization of perfect maps (i.e. continuous closed maps with compact fibers) are presented. Using P λ -spaces and the concept of λ -compactness some classical results regarding λ -perfect maps will be extended. In particular, we show that if the composition f g is a λ -perfect map where f, g are continuous maps with f g well-defined, then f, g are α -perfect and β -perfect, respectively, on appropriate spaces, where α, β ≤ λ .
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