Cardinal inequalities for S(n)-spaces

2018/10/30 by Gotchev, Ivan S. · 1 citation
#54A25 #54D10 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1810.12998

Abstract

Hajnal and Juhász proved that if X is a T₁-space, then |X|≤ 2^s(X)ψ(X), and if X is a Hausdorff space, then |X|≤ 2^c(X)χ(X) and |X|≤ 2^2^s(X). Schröder sharpened the first two estimations by showing that if X is a Hausdorff space, then |X|≤ 2^Us(X)ψ_c(X), and if X is a Urysohn space, then |X|≤ 2^Uc(X)χ(X). In this paper, for any positive integer n and some topological spaces X, we define the cardinal functions χ_n(X), ψ_n(X), s_n(X), and c_n(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.

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• Various S(n)-closednesses in S(n)-spaces with examples

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