In this paper we generalize the notion of of a Tychonoff space to a generic extension of any space by introducing the concept of . This allow us to simplify the treatment in a basic way and in a more general setting. Some [S], [S], and [D]’s results are improved and new characterizations for perfect (Hausdorff) extensions of spaces are obtained.
We show that a recent existence result for the Nash equilibria of generalized games with strategy sets in -spaces is false. We prove that it becomes true if we assume, in addition, that the feasible set of the game (the set of all feasible multistrategies) is closed.
We prove that the maximal Hausdorff compactification of a -compactifiable mapping and the maximal Tychonoff compactification of a Tychonoff mapping (see [P]) are perfect. This allows us to give a characterization of all perfect Hausdorff (respectively, all perfect Tychonoff) compactifications of a -compactifiable (respectively, of a Tychonoff) mapping, which is a generalization of two results of Skljarenko [S] for the Hausdorff compactifications of Tychonoff spaces.
An -space is a topological space in which the topology is generated by the family of all -sets (see [N]). In this paper, minimal--spaces (where denotes several separation axioms) are investigated. Some new characterizations of -spaces are also obtained.
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