A study of continuous selections for filter spaces

设X是一个拓扑空间。我们用2^X表示非空闭子集的集合。具有Vietoris拓扑的集合2 ^X称为超空间。一个映射σ：2^X → X称为选择，如果对每个F ∈ 2^X，σ（F）∈ F.我们刻画了允许连续选择的各种拓扑性质。已知如果2^X容许连续选择，则X是遗传Baire。利用这一事实，我们证明了：（1）可数正则空间允许连续选择当且仅当它是离散的，（2）Hausdorff空间允许Fell连续选择当且仅当它是拓扑良序的，设κ是基数，p是κ上的滤子。我们用κ（p）表示在κ的点处离散的空间，p的邻域基由公式{F ∈ {p}：F ∈ p}给出。对于这类空间，我们得到如下结果：（3）若p有一个套基，则κ（p）允许一个连续选择。(4)ω_1上的余可数滤子允许连续选择，但ω_2上的余可数滤子不允许连续选择。(5)设p_1是κ_1上的基为套的滤子，p_2是ω上的滤子.若κ（p_1）[对称性] ω（p_2）的和容许连续选择，则p_2是Frechet滤子。

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.