No abstract

Abstract. We continue the investigation of star selection principles first considered in [9]. We are concentrated onto star versions of the Hurewicz covering property and star selection principles related to the classes of open covers which have been recently introduced.2000 AMS Classification: 54D20.

The Urysohn number of a space X is U (X) = min{τ : for every subset A ⊂ X such that |A| ≥ τ one can pick neighborhoods Ua ∋ a for all a ∈ A so that ∩ a∈A Ua = ∅}. Some known statements about Urysohn spaces can be generalized in terms of the Urysohn number. (2010): 54A24, 54D10. Mathematics Subject ClassificationKey words: Urysohn space, the Urysohn number of a space, θ-closure, θ-closed hull, the almost Lindelöf degree of a space.

Some variations of Arhangel'skii inequality |X| ≤ 2 χ(X)L(X) for every Hausdorff space X [3], given in [2] and [6] are improved. Mathematics Subject Classification (2010): 54A25, 54D10.

No abstract

A space X is sequentially separable if there is a countable D ⊂ X such that every point of X is the limit of a sequence of points from D. Neither "sequential + separable" nor "sequentially separable" implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered. MSC:54D65, 54A25, 54D55, 54A20 Mikhail "Misha" Matveev passed away unexpectedly on

No abstract

Abstract. We develop a unified framework for the study of classic and new properties involving diagonalizations of dense families in topological spaces. We provide complete classification of these properties. Our classification draws upon a large number of methods and constructions scattered in the literature, and on some novel results concerning the classic properties.