On some classes of Lindelöf Σ-spaces

We consider special subclasses of the class of Lindelöf Σ-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class L Σ ( ⩽ κ ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ ⩽ ω . In the case κ = ω , the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P-approximable spaces considered by Tkačenko. The case κ = 1 corresponds to the spaces of countable network weight, but even the case κ = 2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ 1 is in the class L Σ ( ⩽ ω ) , answering a question of Tkachuk. As well, we study whether certain compact spaces in L Σ ( ⩽ ω ) have dense metrizable subspaces, partially answering a question of Tkačenko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.