On the property of kelley in the hyperspace and Whitney continua

In this paper, we introduce the notion of property K]¹ which implies property K], and we show the following: Let X be a continuum and let ω be any Whitney map for C(X). Then the following are equivalent. (1) X has property K]¹. (2) C(X) has property K]¹. (3) The Whitney continuum ω⁻¹(t) (0⩽t<ω(X)) has property K]¹.As a corollary, we obtain that if a continuum X has property K]¹, then C(X) has property K] and each Whitney continuum in C(X) has property K]. These are partial answers to Nadler's question and Wardle's question (10, (16.37)] and 11, p. 295]).Also, we show that if each continuum X_n (n=1,2,3,…) has property K]¹, then the product ∏X_n has property K]¹, hence C(∏X_n) and each Whitney continuum have property K]¹. It is known that there exists a curve X such that X has property K], but X×X does not have property K] (see 11]).