# Article

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Keywords:
continuum; continuously homogeneous; hyperspace
Summary:
A metric continuum $X$ is said to be continuously homogeneous provided that for every two points $p,q\in X$ there exists a continuous surjective function $f:X\rightarrow X$ such that $f(p)=q$. Answering a question by W.J. Charatonik and Z. Garncarek, in this paper we show a continuum $X$ such that the hyperspace of subcontinua of $X$, $C(X)$, is not continuously homogeneous.
References:
[1] Charatonik J.J., Charatonik W.J.: A degree of nonlocal connectedness. Rocky Mountain J. Math. 31 (2001), 1205–1236. DOI 10.1216/rmjm/1021249438 | MR 1895293
[2] Charatonik W.J., Garncarek Z.: Some remarks on continuously homogeneous continua. Bull. Polish. Acad. Sci. Math. 32 (1984), 339–342. MR 0785993
[3] Engelking R., Lelek A.: Cartesian products and continuous images. Colloq. Math. 8 (1961), 27–29. MR 0131263
[4] Goodykoontz J.T., Jr.: More on connectedness im kleinen and local connectedness in $C(X)$. Proc. Amer. Math. Soc. 65 (1977), 357–364. MR 0451188
[5] Illanes A., Nadler S.B., Jr.: Hyperspaces: Fundamentals and Recent Advances. Monographs and Textbooks in Pure and Applied Mathematics, 216, Marcel Dekker, Inc., New York and Basel, 1999. MR 1670250

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