Cardinal invariants of cellular-Lindelof spaces

Abstract

A space $X$ is said to be "cellular-Lindelöf" if for every cellular family $\mathcal{U}$ there is a Lindelöf subspace $L$ of $X$ which meets every element of $\mathcal{U}$. Cellular-Lindelöf spaces generalize both Lindelöf spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindelöf monotonically normal space is Lindelöf and that every cellular-Lindelöf space with a regular $G_δ$-diagonal has cardinality at most $2^\mathfrak{c}$. We also prove that every normal cellular-Lindelöf first-countable space has cardinality at most continuum under $2^{<\mathfrak{c}}=\mathfrak{c}$ and that every normal cellular Lindelöf space with a $G_δ$-diagonal of rank $2$ has cardinality at most continuum.