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A continuous decomposition of the Menger curve into pseudo-arcs

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It is proved that the Menger universal curve M admits a continuous decomposition into pseudo-arcs with the quotient space homeomorphic to M. Wilson proved [8] Anderson's announcement [1] saying that for any Peano continuum X the Menger universal curve M admits a continuous decomposition into homeomorphic copies of M such that the quotient space is homeomorphic to X. Anderson also announced (unpublished) that the plane admits a continuous decomposition into pseudo-arcs. This result was proved by Lewis and Walsh [4]. In a previous paper [6] the author has proved that each locally planar Peano continuum with no local separating point admits a continuous decomposition into pseudo-arcs. Applying this result, we prove in this note that the Menger universal curve M4 also admits such a decomposition. We can topologically obtain M and some other continua as the quotient space, but not all Peano continua. GENERAL CONSTRUCTIONS AND THEIR PROPERTIES For any compact metric space X let T,I'(X) be the set of all sequences {Xn}, for n E JV = {oo, E, ...}, of closed, mutually disjoint, nonempty subsets of X. Next, let C be the standard Cantor set in the unit interval [0,1]. Fix a sequence of open intervals (an, bn) composed of all, mutually different components of [0, 1] -C. Given a compactum X and a sequence {Xn} E T,J'(X), in the product C x X identify all pairs of points (an, x) and (bn, x), where x E Xn and n E JV. Observe that this identification yields an upper semi-continuous decomposition of C x X. Denote by Q(X, {Xn}) the quotient space of this decomposition and by q the quotient mapping. Property 1. For any compactum X and any sequence {Xn} E T',,(X), we have dimX = dim Q(X, {Xn}). Proof. Letting Q = Q(X, {Xn}), observe that Q contains copies of X, and thus dimQ > dim X. Fix any positive integer n, take the permutation {i1 , ..., in} of {1, ..., n} satisfying 0 < ai1 < bi1 < ... < ain < bin < 1, and define Io = [0,ai1],Ik = [bik,aik+l],If = Received by the editors July 2, 1997 and, in revised form, September 11, 1998. 2000 Mathematics Subject Classification. Primary 54F15, 54F50.