Characterizations of \(\aleph\)-spaces
Two simultaneous generalizations of metrizable spaces and \(\aleph_ 0\)- spaces, namely P. O'Meara's \(\aleph\)-spaces and J. Guthrie's cs-\(\sigma\)- spaces, are shown to be the same. In fact, it is proved that the following are equivalent for a regular space X: (a) X has a \(\sigma\)- locally finite cs-network. (b) X has a \(\sigma\)-discrete k-network. (c) X has a \(\sigma\)-locally finite cs-network. (d) X has a \(\sigma\)-locally finite k-network. [A collection \({\mathcal P}\) of subsets of X is a k-network (resp. cs-network) if, whenever \(S\subset U\) with S compact (resp. S a convergent sequence) and U open in X, then there is \(P\in {\mathcal P}\) such that \(S\subset P\) (resp. S is eventually in P) and \(P\subset U]\).
scientific article; zbMATH DE number 3862006