The hyperspace of finite subsets of a stratifiable space

It is shown that the hyperspace of non-empty finite subsets of a space X is an ANR (an AR) for stratifiable spaces if and only if X is a 2-hyper-locally-connected (and connected) stratifiable space. 0. Introduction. For a space X, let F(X) denote the hyperspace of non-empty finite subsets of X with the Vietoris topology, i.e., the topology generated by the sets 〈U1, . . . , Un〉 = {A ∈ F(X) | A ⊂ U1 ∪ . . .∪Un, Ui ∩A 6= ∅ (∀i = 1, . . . , n)}, where n ∈ N and U1, . . . , Un are open in X. We denote by S the class of stratifiable spaces [Bo1] and byM the class of metrizable spaces. Note that F(X) ∈ S if X ∈ S (cf. [MK, Theorem 3.6]). In [CN], it is shown that F(X) is an ANR(M) (an AR(M)) if and only if X ∈M is locally path-connected (and connected). In this paper, we consider the condition for non-metrizable X ∈ S under which F(X) is an ANR(S) (or an AR(S)). A T1-space X is 2-hyper-locally-connected (2-HLC) [Bo2,3] if there exist a neighborhood U of the diagonal ∆X in X2 and a function λ : U × I→ X satisfying the following conditions: (a) λ(x, y, 0) = x and λ(x, y, 1) = y for each (x, y) ∈ U ; (b) the function t 7→ λ(x, y, t) is continuous for each (x, y) ∈ U ; (c) for each x∈X and each neighborhood V of x, there is a neighborhood W of x such that W 2 ⊂ U and λ(W 2 × I) ⊂ V . The condition (c) means that λ(x, x, t) = x for any x ∈ X and t ∈ I and that λ is continuous at each point of ∆X × I. In case U = X2, X is said to be 2-hyper-connected (2-HC). In the above definition of 2-HLC or 2-HC, 1991 Mathematics Subject Classification: 54B20, 54C55, 54E20.