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Corson compact space; Sokolov space; extent; $\omega $-monolithic space; $\Sigma $-products
We study systematically a class of spaces introduced by Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, $\omega $-stable and $\omega $-monolithic. It is also established that any Sokolov compact space $X$ is Fréchet-Urysohn and the space $C_p(X)$ is Lindelöf. We prove that any Sokolov space with a $G_\delta $-diagonal has a countable network and obtain some cardinality restrictions on subsets of small pseudocharacter lying in $\Sigma $-products of cosmic spaces.
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