## Vladimir V. Tkachuk

*A nice class extracted from $C_p$-theory *

Comment.Math.Univ.Carolinae 46,3 (2005) 503-513. **Abstract:**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 Fr\'echet-Urysohn and the space $C_p(X)$ is Lindel\"of. 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.

**Keywords:** Corson compact space, Sokolov space, extent, $\omega $-monolithic space, $\Sigma $-products

**AMS Subject Classification:** 54B10, 54C05, 54D30

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