## 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