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An example is shown of a continuum X which has no confluent Whitney map for 2X. This answers two problems asked by Nadler [N]. Nadler has asked in [N, (14.63) and (14.64), pp. 468 and 469] whether for every continuum X there exists a monotone (or an open) Whitney map for 2X. We answer both these questions in the negative by showing an example of a continuum X which has no confluent Whitney map for 2X. All considered spaces are assumed to be metric and all mappings are continuous. We denote by R the space of reals and by H the half-line [ 1, + oo). The symbol diam(A) stands for the diameter of a set A. Given a continuum X we use small letters for points of X, capitals to designate subsets of X, and script letters for subsets of 2X. A mapping f: X -Y of a continuum X onto Y is said to be confluent provided that for each subcontinuum K of Y and each component A of f -J(K) we have f(A) = K. It is well known that monotone mappings and open mappings between continua are confluent (cf. [N, (0.45.3), p. 21], where further information is given). THEOREM. There exists a rational continuum X in the plane which admits no confluent Whitney map for 2X. PROOF. Let S denote the unit circle in R2. Define functions f and g mapping H into R2by f(t) = (1 + 1/t)exp(it) and g(t)= (1 l/t)exp(-it), and put M = f(H) and L = g(H). The space X = M U S U L is a rational continuum in R2 (cf. [N, (16.35), p. 558]). We show there is no confluent Whitney map from 2xinto [0, + o) . Let It be an arbitrary Whitney map for 2X. Put = {B E 2X: diam(B) > 1). Thus 2 is a compact subset of 2X. Let to = inf{ (B): B E 2} and note to > 0. Consider the segment [0, t0/2] c [0, IA(X)). We show that there exists a component of -lq([0, t0/2]) whose image under p4 is not the whole segment. To this end put Pk =f(2Tk) E M and qk = g(2Tk) E L for k E {1, 2,. . .}. Observe that the sets { Pk, qk I tend to the one-point set {1} C S, so /({ Pk, qk}) tends to zero as k tends Received by the editors November 21, 1983. 1980 Mathematics Subject Classification. Primary 54B20; Secondary 54C10, 54F20.

A continuum $X$ which has no confluent Whitney map for $2\sp{X}$