ABSTRACT
In [IN, Question 33.10], Illanes and Nadler raised the following question: Is the property of being an hereditarily arcwise connected continuum a sequential strong Whitney-reversible property?
2. PRELIMINARIES
The diameter of a set Am a metric space is denoted by diameter (A). A finite sequence of sets AI,... , An is said to be a weak chain provided that Ai n Aj 7^ 0 if \i — j\ < 1. If X is a continuum, p, is a Whitney map for C (X], and n(A) > t, let C(A,t) = (// | C^A))-1^)- Since C(A,t) is a Whitney level for C(A), C(A,t) is a subcontinuum of ^ (t). Kelley has shown (see [K, Lemma 1.1, p. 23]) that the function a : C (C(X)) -> C (X) defined by a (A) = \J{A : A e A} is continuous and surjective. For x 6 X and e > 0, let Bd(e, x) = {y 6 X : d (x, y) < e}. If X is a continuum and B C -X", then let JV(e, J3) = IJ{^ (e 'x) : x e £?}. A continuous function u; : C (X) —> [0, oo) is a sue map provided that u ({x}) = 0 and, if A C 5, u; (A) < u(B}. An order arc in C(X) is an arc a in C (X) such that if A, B € a, then A C B or B C A By Theorem 1.8 of [Nl], for each A, B e C(X) such that A C B and .A ^ 5, there is an order arc joining A and B.
