Star countable spaces and $$\omega $$ ω -domination of discrete subspaces

Abstract

We establish that a first countable $$\omega $$ -monolithic space is star countable if and only if it has countable extent. A consistent example is given of a first countable normal star Lindelof space of uncountable extent. Under the continuum hypothesis we prove that for any compact K, the space $$C_p(K)$$ is star countable if and only if it is Lindelof. The above-mentioned results answer several published open questions.