Function space topologies are investigated for the class of continuous multifunctions. Using the notion of continuous convergence, splittingness and admissibility are discussed for the topologies on continuous multifunctions. The theory of net of sets is further developed for this purpose. The (τ, µ)-topology on the class of continuous multifunctions is found to be upper admissible, while the compact-open topology is upper splitting. The point-open topology is the coarsest topology which is coordinately admissible, it is also the finest topology which is coordinately splitting.2010 MSC: 54C35; 54A05; 54C60.

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We define and study a new class of regular sets calledPS-regular sets. Properties of these sets are investigated for topological spaces and generalized topological spaces. Decompositions of regular open sets and regular closed sets are provided usingPS-regular sets. Semiconnectedness is characterized by usingPS-regular sets.PS-continuity and almostPS-continuity are introduced and investigated.

Abstract. If X is a topological space and A ⊆ X, then the number of distinct sets that can be obtained from A by using all possible compositions for operators iγ , cγ (where γ = σ, π, α, β) introduced by Császár is at the most 25. Explicit expressions for these sets are provided. An example is provided where all the 25 different sets are determined. The result is also discussed for special cases such as when the space is extremally disconnected, resolvable, open-unresolvable, and partition spaces.

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