On the cardinality of Urysohn spaces and weakly $H$-closed spaces

Abstract

We introduce the cardinal invariant $\theta$-$aL'(X)$, related to $\theta$-$aL(X)$, and show that if $X$ is Urysohn, then $|X|\leq2^{\theta\text-aL'(X)\chi(X)}$. As $\theta$-$aL'(X)\leq aL(X)$, this represents an improvement of the Bella-Cammaroto inequality. We also introduce the classes of firmly Urysohn spaces, related to Urysohn spaces, strongly semiregular spaces, related to semiregular spaces, and weakly $H$-closed spaces, related to $H$-closed spaces.

Keywords

• Urysohn space
• $\theta$-closure
• pseudocharacter
• almost Lindelöf degree
• cardinality
• cardinal invariant