Relative compactness and networks (Q1358589)

It is known [\textit{A. V. Arkhangel'skij}, Dokl. Akad. Nauk SSSR 126, 239-241 (1959; Zbl 0087.37602)] that if in a compact Hausdorff space a net of cardinality \(\tau\) exists, then in this space also a base of cardinality \(\tau\) exists. In the present paper this result is extended to the case of relative compactness. Theorem 1 obtained in this direction is the main result of this paper: If \(Y\) is compact in the Hausdorff space \(X\) and the space \(X\) has a countable net, then in the space \(Y\) a countable base exists. The paper also contains some applications of Theorem 1 and an example demonstrating its finality.