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.

In this paper some cardinal inequalities for Urysohn spaces are established. In particular the following two theorems are proved:(i)If where [A]θ denotes the θ-closed hull of A, i.e., the smallest θ-closed subset of X containing A;(ii), where aL(X, X) is the smallest cardinal number m such that for every open cover of X there is a subfamily for which

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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.

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