Ivan Gotchev

Cardinal inequalities for $S(n)$ -spaces.

The $S(n)$ -spaces, where $n$ is a positive integer, were introduced in 1969 by Viglino who showed that for every $n$ there is an $S(n)$ -space which is not $S(n+1)$ , and that there are spaces which are $S(n)$ for every $n$ , but are not regular. We recall that the $S(1)$ -spaces are exactly the Hausdorff spaces and $S(2)$ -spaces are the Urysohn spaces.

In 1973, Porter and Votaw, extended Viglino’s definition to every infinite ordinal number $\alpha$ and characterized the minimal $S(\alpha)$ and $S(\alpha)$ -closed spaces, and in 1988, Dikranjan and Giuli characterized the $S(n)$ – $\theta$ -closed spaces and Dikranjan, Giuli and Tholen proved that the category of $S(n)$ -spaces is not cowellpowered for every $n>1$ .

In this talk we will show how by using new cardinal invariants defined specifically for $S(n)$ -spaces, some well-known cardinal inequalities of Pospišil ( $|X|\le 2^{2^{d(X)}}$ and $|X|\le d(X)^{\chi(X)}$ ), Hajnal and Juhász ( $|X| \le2^{c(X)\chi(X)}$ and $|X| \le 2^{2^{s(X)}}$ ), Arhangel’skiı̆ ( $|X|\le d(X)^{bt(X)}$ ), and others, proved for Hausdorff spaces, could be sharpened for $S(n)$ -spaces.

Algebra, Topology, and Dynamical Systems

A conference in honor of Dikran Dikranjan on the occasion of his 75th birthday and of Ivan Prodanov on the occasion of the 90th anniversary of his birth

Ivan Gotchev