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]).