Continuously homogeneous continua and their arc components

Author:
Janusz R. Prajs

Journal:
Proc. Amer. Math. Soc. **101** (1987), 533-540

MSC:
Primary 54F20; Secondary 54F65

MathSciNet review:
908664

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a continuously homogeneous Hausdorff continuum. We prove that if there is a sequence of its arc components with , and there is an arc component of with nonempty interior, then is arcwise connected. As consequences and applications we get: (1) if is the countable union of arcwise connected continua, then is arcwise connected; (2) if is nondegenerate and metric, the number of its arc components is countable and it contains no simple triod, then it is either an arc or a simple closed curve; and, in particular, (3) an arc is the only nondegenerate continuously homogeneous arc-like metric continuum with countably many arc components.

**[C]**Janusz J. Charatonik,*A characterization of the pseudo-arc*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**26**(1978), no. 11, 901–903 (1979) (English, with Russian summary). MR**524921****[CM]**J. J. Charatonik and T. Maćkowiak,*Confluent and related mappings on arc-like continua—an application to homogeneity*, Topology Appl.**23**(1986), no. 1, 29–39. MR**849092**, 10.1016/0166-8641(86)90015-5**[CG]**Włodzimierz J. Charatonik and Zbigniew Garncarek,*Some remarks on continuously homogeneous continua*, Bull. Polish Acad. Sci. Math.**32**(1984), no. 5-6, 339–342 (English, with Russian summary). MR**785993****[H1]**Charles L. Hagopian,*Mapping theorems for plane continua*, Proceedings of the 1978 Topology Conference (Univ. Oklahoma, Norman, Okla., 1978), I, 1978, pp. 117–122 (1979). MR**540483****[H2]**Charles L. Hagopian,*Planar 𝜆 connected continua*, Proc. Amer. Math. Soc.**39**(1973), 190–194. MR**0315681**, 10.1090/S0002-9939-1973-0315681-4**[H3]**Charles L. Hagopian,*𝜆 connected plane continua*, Trans. Amer. Math. Soc.**191**(1974), 277–287. MR**0341435**, 10.1090/S0002-9947-1974-0341435-4**[K1]**P. Krupski,*Continua which are homogeneous with respect to continuity*, Houston J. Math.**5**(1979), no. 3, 345–356. MR**559975****[K2]**P. Krupski,*Continuous homogeneity of continua*, Proceedings of the International Conference on Geometric Topology (Warsaw, 1978) PWN, Warsaw, 1980, pp. 269–272. MR**656755****[KM]**K. Kuratowski and A. Mostowski,*Set theory*, PWN-Polish Scientific Publishers, Warszawa, 1967.**[L]**A. Lelek,*On weakly chainable continua*, Fund. Math.**51**(1962/1963), 271–282. MR**0143182****[P1]**Janusz R. Prajs,*On continuous mappings onto some Cartesian products of compacta*, Houston J. Math.**14**(1988), no. 4, 573–582. MR**998458****[P2]**-,*Some invariants under perfect mappings and their applications to continua*(to appear).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
54F20,
54F65

Retrieve articles in all journals with MSC: 54F20, 54F65

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0908664-4

Keywords:
Continuous homogeneity,
covering sequence,
Hausdorff continuum,
-component,
simple triod

Article copyright:
© Copyright 1987
American Mathematical Society