On some covering properties of metric spaces
We prove a number of results on star covering properties which may be regarded as either generalizations or specializations of topological properties related to the ones mentioned in the title of the paper. For instance, we give a new, entirely combinatorial proof of the fact that -spaces constructed from infinite almost disjoint families are not star-compact. By going a little further we conclude that if is a star-compact space within a certain class, then is neither first countable nor separable....
A space is -starcompact if for every open cover of there exists a Lindelöf subset of such that We clarify the relations between -starcompact spaces and other related spaces and investigate topological properties of -starcompact spaces. A question of Hiremath is answered.
A space is -starcompact if for every open cover of there exists a countably compact subset of such that In this paper we investigate the relations between -starcompact spaces and other related spaces, and also study topological properties of -starcompact spaces.
We introduce the cardinal invariant -, related to -, and show that if is Urysohn, then . As -, 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 -closed spaces, related to -closed spaces.