We prove the following results. (i) Let X be a continuum such that X contains a dense arc component and let D be a dendrite with a closed set of branch points. If f:X→D is a Whitney preserving map, then f is a homeomorphism. (ii) For each dendrite D′ with a dense set of branch points there exist a continuum X′ containing a dense arc component and a Whitney preserving map f′:X′→D′ such that f′ is not a homeomorphism.
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We prove that the set of all Krasinkiewicz maps from a compact metric space to a polyhedron (or a 1-dimensional locally connected continuum, or an n-dimensional Menger manifold, n ≥ 1) is a dense Gδ-subset of the space of all maps. We also investigate the existence of surjective Krasinkiewicz maps from continua to polyhedra.
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In [7], M. Levin proved that the set of all Bing maps of a compact metric space to the unit interval is a dense G[delta]-subset of the space of all maps. In [6], J. Krasinkiewicz . independently proved that the set of all Bing maps of a compact metric space to an n-dimensional manifold (n is less than or equal to 1) is a dense G[delta]-subset of the space of maps. In [9], J. Song and E. D. Tymchatyn, solving some problems of J. Krasinkiewicz ([6]), proved that the set of all Bing maps of a compact metric space to a nondegenerate connected polyhedron is a dense G[delta]-subset of the space of maps. In this note, we investigate the existence of surjective Bing maps from continua to polyhedra.
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