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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.

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