No homogeneous tree-like continuum contains an arc
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- by Charles L. Hagopian PDF
- Proc. Amer. Math. Soc. 88 (1983), 560-564 Request permission
Abstract:
In 1960 R. H. Bing [4, p. 210] asked the following question. "Is there a homogeneous tree-like continuum that contains an arc?" We answer this question in the negative. This result generalizes Bingβs theorem [4, p. 229] that every atriodic homogeneous tree-like continuum is arcless.References
- R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729β742. MR 27144
- R. H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653β663. MR 43450
- R. H. Bing, Each homogeneous nondegenerate chainable continuum is a pseudo-arc, Proc. Amer. Math. Soc. 10 (1959), 345β346. MR 105072, DOI 10.1090/S0002-9939-1959-0105072-6
- R. H. Bing, A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Canadian J. Math. 12 (1960), 209β230. MR 111001, DOI 10.4153/CJM-1960-018-x
- R. H. Bing, The elusive fixed point property, Amer. Math. Monthly 76 (1969), 119β132. MR 236908, DOI 10.2307/2317258
- C. E. Burgess, Chainable continua and indecomposability, Pacific J. Math. 9 (1959), 653β659. MR 110999
- C. E. Burgess, Homogeneous continua which are almost chainable, Canadian J. Math. 13 (1961), 519β528. MR 126255, DOI 10.4153/CJM-1961-043-8 β, Homogeneous $1$-dimensional continua, General Topology and Modern Analysis, Academic Press, New York, 1981, pp. 169-175.
- Edward G. Effros, Transformation groups and $C^{\ast }$-algebras, Ann. of Math. (2) 81 (1965), 38β55. MR 174987, DOI 10.2307/1970381
- Charles L. Hagopian, Homogeneous plane continua, Houston J. Math. 1 (1975), 35β41. MR 383369
- Charles L. Hagopian, Indecomposable homogeneous plane continua are hereditarily indecomposable, Trans. Amer. Math. Soc. 224 (1976), no.Β 2, 339β350 (1977). MR 420572, DOI 10.1090/S0002-9947-1976-0420572-1
- Charles L. Hagopian, A characterization of solenoids, Pacific J. Math. 68 (1977), no.Β 2, 425β435. MR 458381
- Charles L. Hagopian, Atriodic homogeneous continua, Pacific J. Math. 113 (1984), no.Β 2, 333β347. MR 749539
- F. Burton Jones, Certain homogeneous unicoherent indecomposable continua, Proc. Amer. Math. Soc. 2 (1951), 855β859. MR 45372, DOI 10.1090/S0002-9939-1951-0045372-4
- Wayne Lewis, Almost chainable homogeneous continua are chainable, Houston J. Math. 7 (1981), no.Β 3, 373β377. MR 640978 β, The pseudo-arc of pseudo-arcs is unique, preprint. R. L. Moore, Concerning triodic continua in the plane, Fund. Math. 13 (1929), 261-263. J. T. Rogers, Jr., Homogeneous, hereditarily indecomposable continua are tree-like, preprint.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 88 (1983), 560-564
- MSC: Primary 54F15; Secondary 54F20
- DOI: https://doi.org/10.1090/S0002-9939-1983-0699434-X
- MathSciNet review: 699434