Cardinal inequalities for \(S(n)\)-spaces (Q2330070)

Recall that for \(n\in\mathbb{N}\) a space \(X\) is an \(S(n)\)-space provided that for any two points \(x,y\in X\) (\(x\not=y\)) there are open sets \(\langle U_i\rangle_{i=1,\dots,n}\) with \(x\in U_1\), \(\overline{U_i}\subset U_{i+1}\) and \(y\notin\overline{U_n}\). Cardinal functions \(\chi_n(X)\) (\(S(n)\)-character), \(\psi_n(X)\) (\(S(n)\)-pseudocharacter), \(s_n(X)\) (\(S(n)\)-spread) and \(c_n(X)\) (\(S(n)\)-cellularity) are defined and then they are used to prove a number of bounds on the cardinality of \(S(n)\)-spaces related to known cardinal inequalities for \(T_1\), \(T_2\) and Urysohn spaces, for example if \(X\) is an \(S(n)\)-space then \(\vert X\vert \le 2^{c_n(X)\chi_n(X)}\) and \(\vert X\vert \le 2^{2^{s_{2k}(X)}}\) when \(X\) is an \(S(3k)\)-space while \(\vert X\vert \le 2^{2^{s_{2k-1}(X)}}\) when \(X\) is an \(S(3k-2)\)-space.