THE SPAN OF HYPERSPACES HIROSHI HOSOKAWA

Abstract. Let X be a continuum, C(X) the hyperspace of X, µ: C(X)→ [0, 1] a Whitney map, t ∈ [0, 1) and let τ be one of the spans σ∗, σ∗0, σ and σ0. Then the followings hold: (1) τ(C(X))> 0; (2) if τ(µ−1(t)) = 0, then τ(µ−1(s)) = 0 for all s ≥ t. In particular, the property of having zero span is a Whitney property; (3) the function F: [0, 1] → [0,diam X] defined by F (t) = τ(µ−1(t)) is continuous. 1. Preliminary In this paper, continua are nonempty compact connected metric spaces and mappings are continuous functions. The hyperspace of a continuum X is the space C(X) consisting of all nonempty subcontinua of X with the Hausdorff metric H defined by H(K,L) = inf{ε> 0: L ⊂ N(K, ε) and K ⊂ N(L, ε)} for each K, L ∈ C(X), where for any subset A of X and any positive number ε, N(A, ε) is the open ε-neighborhood around A in X. It is known that C(X) is a continuum (see [5], Theorem 2.7). A Whitney map µ: C(X) → [0, 1] is a mapping such that µ({x}) = 0 for each x ∈ X, µ(X) = 1 and if K, L ∈ C(X) with K ⊂ L 6 = K, then µ(K) < µ(L). Such a mapping always exists for any nondegenerate continuum X (see [10] or [11]). The subspace µ−1(t) of C(X) is called a Whitney level for any Whitney map µ: C(X) → [0, 1] and for any t ∈ [0, 1). It is known that µ−1([0, t]), µ−1([t, 1]) and µ−1(t) are connected. More precisely, every Whitney map for C(X) is monotone and open (see [10], Theorem 14.2 and 1.213.1). A.Lelek defined spans of metric spaces ([7] and [8])