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 LΣ(⩽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 ℵ1 is in the class LΣ(⩽ω), answering a question of Tkachuk. As well, we study whether certain compact spaces in LΣ(⩽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.