On some classes of Lindelöf Sigma-spaces

Abstract

We consider special subclasses of the class of Lindelöf Sigma-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 $L\Sigma (\le \kappa )$ if it admits a cover by compact subspaces of weight $\kappa$ and a countable network for the cover. We restrict our attention to $\kappa \le \omega$. In the case $\kappa =\omega$, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the $P$-approximable spaces considered by Tkacenko. The case $\kappa =1$ corresponds to the spaces of countable network weight, but even the case $\kappa =2$ 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 ${\aleph }_1$ is in the class $L\Sigma (\le \omega )$, answering a question of Tkachuk. As well, we study whether certain compact spaces in $L\Sigma (\le \omega )$ have dense metrizable subspaces, partially answering a question of Tkacenko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.

21 pages. to appear in Topology and its Applications

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