On open-separated sequences

Abstract

We introduce the notion of an open-separated sequence, related to a free sequence, and define the cardinal function \(\text{os} ( X)\). Nonregular and regular examples \(X\) are given such that \(\text{os} ( X)<F(X)\). Motivated by a question of Angelo Bella, we show that if \(X\) is Hausdorff then \(|X|\leq hL(X)^{\text{os} ( X)\psi_c(X)}\). As \(\text{os} ( X)\psi_c(X)\leq hL(X)\) if \(X\) is Hausdorff, this gives a strengthening of the De Groot-Smirnov bound \(2^{hL(X)}\) for the cardinality of a Hausdorff space. Additionally we show \( \text{nw}( X)\leq \text{hL}(X)^{\text {os} ( X)}\) if \(X\) is regular. A consequence is that if \(X\) is regular and either almost radial or hereditarily weakly Whyburn then \( { |X|\leq hL(X)^{\text{os} ( X)} } \).