A study of continuous selections for filter spaces

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Let X be a topological space. We denote by 2^X the collection of non-empty closed subsets. The set 2^X with the Vietoris topologyis called hyperspace. A map σ : 2^X → X is called a selection if σ(F) ∈ F for every F ∈ 2^X. We characterize various topological properties which admit continuous selections. It is known that if 2^X admits a continuous selection, then X is hereditarily Baire. Using this fact we have shown:(1) A countable regular space admits a continuous selection if and only if it is scattered.Also we have shown:(2) A Hausdorff space admits a Fell continuous selection if and only if it is topological well-orderable.Let κ be cardinal and let p be a filter on κ. By κ(p) we denote the space which is discrete at points of κ and a neighborhood base of p is given by the fomular {F ∪ {p} : F ∈ p}. For these type of spaces we have the following results:(3) If p has a nested base, then κ(p) admits a continuous selection.(4) A co-countable filter on ω_1 admits a continuous selection but not on ω_2.(5) Let p_1 be a filter on κ_1 with a nested base and p_2 be a filter on ω. If the sum of κ(p_1) 【symmetry】 ω(p_2) admits a continuous selection, then p_2 is the Frechet filter.