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On problems of H. Cook. (English) Zbl 0612.54042
Let $${\mathcal F}$$ be a class of mapping such that all homeomorphisms are in $${\mathcal F}$$ and the composition of any two mappings in $${\mathcal F}$$ is also in $${\mathcal F}$$. In University of Houston Mathematics Problem Book (Problem 150 and Problem 151), H. Cook has the following problems: (i) If a continuum X is homogeneous with respect to $${\mathcal F}$$, is there a continuum Y which is $${\mathcal F}$$-equivalent to X and which is homogeneous? (ii) Suppose that a continuum Y is $${\mathcal F}$$-equivalent to a homogeneous continuum and $$\epsilon$$ is a positive number. Does there exist a positive number $$\delta$$ such that if a,b$$\in Y$$ and $$d(a,b)<\delta$$, then there is a mapping f in $${\mathcal F}$$ from Y onto Y such that $$f(a)=b$$ and no point of Y is moved a distance more than $$\epsilon$$ ? We give negative answers to the problems (i) and (ii).

##### MSC:
 54F15 Continua and generalizations 54C05 Continuous maps 54F50 Topological spaces of dimension $$\leq 1$$; curves, dendrites 54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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##### References:
 [1] Charatonik, J.J., A characterization of the pseudo-arc, Bull. Pol. acad. sci. ser. math., 26, 901-903, (1978) · Zbl 0433.54025 [2] Charatonik, J.J., Generalized homogeneity and some characterizations of solenoids, Bull. Pol. acad. sci. ser. math., 31, 171-174, (1983) · Zbl 0529.54032 [3] Charatonik, J.J., The property of Kelley and confluent mappings, Bull. Pol. acad. sci. ser. math., 31, 375-380, (1983) · Zbl 0544.54028 [4] Charatonik, J.J., Mappings of sierpinski curve onto itself, Proc. amer. math. soc., 92, 125-132, (1984) · Zbl 0524.54010 [5] Effros, E.G., Transformation groups and C∗-algebra, Ann. of math., 81, 2, 38-55, (1965) · Zbl 0152.33203 [6] Hu, S.T., Theory of retracts, (1965), Wayne State Univ. Press Detroit · Zbl 0137.01701 [7] H. Kato, Generalized homogeneity of continua and a question of J.J. Charatonik, Houston J. Math., to appear. · Zbl 0635.54017 [8] Krupski, P., Continua which are homogeneous with respect to continuity, Houston J. math., 5, 345-355, (1979) · Zbl 0415.54019 [9] Kuratowski, K., Topology, Vol. II, (1968), Academic Press New York, PWN, Warsaw [10] McLean, T.B., Confluent images of tree-like curves are tree-like, Duke math. J., 39, 465-473, (1972) · Zbl 0252.54020 [11] Taylor, J.L., A counterexample in shape theory, Bull. amer. math. soc., 81, 629-632, (1975) · Zbl 0316.55010
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