A PROPERTY BETWEEN COMPACT AND STRONGLY COUNTABLY COMPACT

Dusan Milovancevi\'c

Abstract: In this paper we consider a class of spaces called hypercountably compact (hcc)spaces. The class of countably compact and the class of strongly countably compact (scc) spaces contain the class of hypercountably compact spaces. In example 2.1, we give a strongly countably compact space which is not hypercountably compact. In the class of spaces satisfying the first axiom of countability the notions hcc and scc coincide (Theorem 2.3). Some equivalent conditions for a space to be hcc are given in Theorem 2.2. The hcc property is not a continuous invariant (Example 2.4). In section 3 we consider compact spaces which contain noncompact hcc (scc) spaces as subspaces. In section 4 we also consider strongly sequentially compact (ssc) spaces.

Classification (MSC2000): 54D30, 54820

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