Some examples concerning $L\Sigma(\leq\omega)$ and metrizably fibered compacta

The class of $L\Sigma(\leq\omega)$-spaces was introduced in 2006 by Kubiś, Okunev and Szeptycki as a natural refinement of the classical and important notion of Lindelöf $\Sigma$-spaces. Compact $L\Sigma(\leq\omega)$-spaces were considered earlier, under different names, in the works of Tkachuk and Tkachenko in relation to metrizably fibered compacta. In this paper we give counterexamples to several open questions about compact $L\Sigma(\leq\omega)$-spaces that are scattered in the literature. Among other things, we refute a conjecture of Kubiś, Okunev and Szeptycki by constructing a separable Rosenthal compactum which is not an $L\Sigma(\leq\omega)$-space. We also give insight to the structure of first-countable $(K)L\Sigma(\leq\omega)$-compacta.