On o-free sequences and compacta

Abstract

We use the notion of an o-free sequence to give new bounds for the cardinality of Hausdorff spaces and regular spaces. There are several implications for compacta. One is that if X is a compactum then $w\left(X\right)\le hL{\left(X\right)}^{ot\left(X\right)}$, where $ot\left(X\right)$ is the o-tightness introduced by Tkachenko. Another is that $\left|X\right|\le hL{\left(X\right)}^{ot\left(X\right)w{\psi }_c\left(X\right)}$ if X is a compactum. This is shown to be a strict improvement of Arhangel'skiĭ's bound $2^{\psi \left(X\right)}$. Finally, we show $\left|X\right|\le hL{\left(X\right)}^{ot\left(X\right)\pi \chi \left(X\right)}$ if X is a homogeneous compactum. We note $hL{\left(X\right)}^{ot\left(X\right)\pi \chi \left(X\right)}\le 2^{t\left(X\right)}$ for such spaces, where $2^{t\left(X\right)}$ is de la Vega's bound for the cardinality of homogeneous compacta.