Homogeneous continua in Euclidean -space which contain an -cube are -manifolds

Author:
Janusz R. Prajs

Journal:
Trans. Amer. Math. Soc. **318** (1990), 143-148

MSC:
Primary 54F20; Secondary 57N35

MathSciNet review:
943307

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Abstract: Let be a homogeneous continuum and let be Euclidean -space. We prove that if is properly contained in a connected -manifold, then contains no -dimensional umbrella (i.e. a set homeomorphic to the set and and either or ). Combining this fact with an earlier result of the author we conclude that if lies in and topologically contains , then is an -manifold.

**[B]**R. H. Bing,*A simple closed curve is the only homogeneous bounded plane continuum that contains an arc*, Canad. J. Math.**12**(1960), 209–230. MR**0111001****[Bo]**K. Borsuk,*Theorem of retracts*, PWN, Warsaw, 1967.**[H]**Charles L. Hagopian,*Homogeneous plane continua*, Houston J. Math.**1**(1975), 35–41. MR**0383369****[M]**S. Mazurkiewicz,*Sur les continus homogènes*, Fund. Math.**5**(1924), 137-146.**[P]**Janusz R. Prajs,*Homogeneous continua in Euclidean (𝑛+1)-space which contain an 𝑛-cube are locally connected*, Trans. Amer. Math. Soc.**307**(1988), no. 1, 383–394. MR**936823**, 10.1090/S0002-9947-1988-0936823-9

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-0943307-X

Keywords:
Continuum,
Euclidean space,
homogeneity,
-dimensional umbrella,
-manifold

Article copyright:
© Copyright 1990
American Mathematical Society