The well-known result of S. Mazurkiewicz that the simple closed curve is the only nondegenerate locally connected plane homogeneous continuum is extended to generalized homogeneity with respect to some other classes of mappings. Several open problems in the area are posed.
@article{118457, author = {Janusz Jerzy Charatonik}, title = {On generalized homogeneity of locally connected plane continua}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {32}, year = {1991}, pages = {769-774}, zbl = {0786.54035}, mrnumber = {1159824}, language = {en}, url = {http://dml.mathdoc.fr/item/118457} }
Charatonik, Janusz Jerzy. On generalized homogeneity of locally connected plane continua. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) pp. 769-774. http://gdmltest.u-ga.fr/item/118457/
A characterization of the universal curve and a proof of its homogeneity, Ann. of Math. (2) 67 (1958), 313-324. (1958) | MR 0096180 | Zbl 0083.17607
One-dimensional continuous curves and a homogeneity theorem, Ann. of Math. (2) 68 (1958), 1-16. (1958) | MR 0096181 | Zbl 0083.17608
A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Canad. J. Math. 12 (1960), 209-230. (1960) | MR 0111001 | Zbl 0091.36204
On fans, Dissertationes Math. (Rozprawy Mat.) 54 (1967), 1-40. (1967) | MR 0227944 | Zbl 0163.44604
Some problems on generalized homogeneity of continua, Topology, Proc. International Topological Conference, Leningrad 1982; Lecture Notes in Math., vol. 1060, Springer Verlag, 1984, 1-6. | MR 0770218 | Zbl 0544.54029
Mappings of the Sierpiński curve onto itself, Proc. Amer. Math. Soc. 92 (1984), 125-132. (1984) | MR 0749904 | Zbl 0524.54010
Generalized homogeneity of curves and a question of H. Kato, Bull. Polish Acad. Sci. Math. 36 (1988), 409-411. (1988) | MR 1101430 | Zbl 0767.54030
Monotone mappings of universal dendrites, Topology Appl. 38 (1991), 163-167. (1991) | MR 1094549 | Zbl 0726.54012
On light open mappings, Proceedings International Topological Conference, Baku (USSR), l989, 211-219. | MR 1347226 | Zbl 0817.54010
Generalized homogeneity of continua and a question of J.J. Charatonik, Houston J. Math. 13 (1987), 51-63. (1987) | MR 0884233 | Zbl 0635.54017
On problems of H. Cook, Topology Appl. 26 (1987), 219-228. (1987) | MR 0904468 | Zbl 0612.54042
On homeomorphisms of the Sierpiński curve, Comment. Math. Prace Mat. 12 (1969), 255-257. (1969) | MR 0247618 | Zbl 0235.54039
Sur les continus homogènes, Fund. Math. 5 (1924), 137-146. (1924)
On open homogeneity of closed balls in the Euclidean spaces, preprint.
Openly homogeneous continua in 2-manifolds, preprint. | Zbl 0830.54028
Homogeneous continua, Topology Proc. 8 (1983), 213-233. (1983) | MR 0738476 | Zbl 0541.54039
Classifying homogeneous continua, Proc. of the Topology Symposium at Oxford University (June 1989), preprint. | MR 1173271 | Zbl 0776.54023
Open mappings of the universal curve onto continuous curves, Trans. Amer. Math. Soc. 168 (1972), 497-515. (1972) | MR 0298630 | Zbl 0239.54007