On homogeneous hereditarily unicoherent continua
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- by G. R. Gordh PDF
- Proc. Amer. Math. Soc. 51 (1975), 198-202 Request permission
Abstract:
Let $\mathfrak {M}$ denote the class of all hereditarily unicoherent Hausdorff continua in which each indecomposable subcontinuum is irreducible. It is shown that if the continuum $M$ in $\mathfrak {M}$ is decomposable, then the set of weak terminal points of $M$ is a nonempty, proper subset. The following generalization of a theorem of F. Burton Jones is an immediate corollary: if the continuum $M$ in $\mathfrak {M}$ is homogeneous, then $M$ is indecomposable. As an application, it is proved that if $X$ is a homogenous, hereditarily unicoherent Hausdorff continuum which is an image of an ordered compactum, then $X$ is an indecomposable metrizable continuum.References
- David P. Bellamy, Composants of Hausdorff indecomposable continua; a mapping approach, Pacific J. Math. 47 (1973), 303–309. MR 331345
- G. R. Gordh Jr., Monotone decompositions of irreducible Hausdorff continua, Pacific J. Math. 36 (1971), 647–658. MR 281163 —, Indecomposable Hausdorff continua and mappings of connected linearly ordered spaces, Proc. Univ. of Oklahoma Conf. on General Topology, 1972.
- G. R. Gordh Jr., Terminal subcontinua of hereditarily unicoherent continua, Pacific J. Math. 47 (1973), 457–464. MR 362268
- G. R. Gordh Jr., Indecomposable Hausdorff continua and mappings of connected linearly ordered spaces, Glasnik Mat. Ser. III 9(29) (1974), 137–139 (English, with Serbo-Croatian summary). MR 350710
- John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
- F. Burton Jones, Certain homogeneous unicoherent indecomposable continua, Proc. Amer. Math. Soc. 2 (1951), 855–859. MR 45372, DOI 10.1090/S0002-9939-1951-0045372-4
- F. Burton Jones, Homogeneous plane continua, Proceedings of the Auburn Topology Conference (Auburn Univ., Auburn, Ala., 1969; dedicated to F. Burton Jones on the occasion of his 60th birthday), Auburn Univ., Auburn, Ala., 1969, pp. 46–56. MR 0391040 S. Mardešić, On the Hahn-Mazurkiewicz problem in nonmetric spaces, Proc. Second Prague Topological Sympos., 1966, pp. 248-255.
- Harlan C. Miller, On unicoherent continua, Trans. Amer. Math. Soc. 69 (1950), 179–194. MR 36498, DOI 10.1090/S0002-9947-1950-0036498-3
- L. B. Treybig, Concerning homogeneity in totally ordered, connected topological space, Pacific J. Math. 13 (1963), 1417–1421. MR 159309
- L. B. Treybig, Concerning continua which are continuous images of compact ordered spaces, Duke Math. J. 32 (1965), 417–422. MR 187220
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 51 (1975), 198-202
- MSC: Primary 54F20
- DOI: https://doi.org/10.1090/S0002-9939-1975-0375254-6
- MathSciNet review: 0375254