Cardinal inequalities for S(n)-spaces

Abstract

Hajnal and Juhasz [9] proved that if X is a T1-space, then $${|X| \leq 2^{s(X)\psi(X)}}$$ , and if X is a Hausdorff space, then $${|X| \leq 2^{c(X)\chi(X)}}$$ and $${|X| \leq 2^{2^{s(X)}}}$$ . Schroder sharpened the first two estimations by showing that if X is a Hausdorff space, then $${|X| \leq 2^{Us(X)\psi_c(X)}}$$ , and if X is a Urysohn space, then $${|X| \leq 2^{Uc(X)\chi(X)}}$$ . In this paper, for any positive integer n and some topological spaces X, we define the cardinal functions $${\chi_n(X), \psi_n(X), s_n(X)}$$ , and cn(X) called respectively S(n)-character, S(n)-pseudocharacter, S(n)-spread, and S(n)-cellularity and using these new cardinal functions we show that the above-mentioned inequalities could be extended to the class of S(n)-spaces. We recall that the S(1)-spaces are exactly the Hausdorff spaces and the S(2)-spaces are exactly the Urysohn spaces.