On the upper bound of the cardinality of Hausdorff topological spaces

Abstract

Let X be a Hausdorff space. In 1967, Hajnal and Juhász, and in 1969, Arhangel′skiĭ proved, respectively, that |X|≤2χ(X)c(X) and |X|≤2χ(X)L(X). Using Pospišil's inequality |X|≤d(X)χ(X), Pol gave short proofs of these two inequalities in 1974. In 1988, Sun generalized Hajnal–Juhász' inequality by showing that |X|≤πχ(X)c(X)ψc(X). Also, in 1984, Willard and Dissanayake improved Pospišil's inequality by showing that |X|≤d(X)πχ(X)ψc(X) and in 2016, Gotchev, Tkachenko and Tkachuk improved Pospišil's inequality and Sun's inequality by showing that |X|≤πw(X)ot(X)ψc(X).In this paper we prove that |X|≤πw(X)dot(X)ψc(X) and we show that this upper bound is either the same or less than the upper bound given by any one of the above-mentioned inequalities.