The local structure of homogeneous continua (curves) is studied. Components of open subsets of each homogeneous curve which is not a solenoid have the disjoint arcs property. If the curve is aposyndetic, then the components are nonplanar. A new characterization of solenoids is formulated: a continuum is a solenoid if and only if it is homogeneous, contains no terminal nontrivial subcontinua and small subcontinua are not ∞-ods.
@article{bwmeta1.element.bwnjournal-article-fmv146i2p159bwm,
author = {Pawe\l\ Krupski},
title = {The disjoint arcs property for homogeneous curves},
journal = {Fundamenta Mathematicae},
volume = {146},
year = {1995},
pages = {159-169},
zbl = {0831.54031},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv146i2p159bwm}
}
Krupski, Paweł. The disjoint arcs property for homogeneous curves. Fundamenta Mathematicae, Tome 146 (1995) pp. 159-169. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv146i2p159bwm/
[00000] [1] R. D. Anderson, One-dimensional continuous curves and a homogeneity theorem, Ann. of Math. 68 (1958), 1-16. | Zbl 0083.17608
[00001] [2] M. Bestvina, Characterizing k-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 380 (1988). | Zbl 0645.54029
[00002] [3] J. H. Case, Another 1-dimensional homogeneous continuum which contains an arc, Pacific J. Math. 11 (1961), 455-469. | Zbl 0101.15404
[00003] [4] E. Duda, P. Krupski and J. T. Rogers, On locally chainable homogeneous continua, Topology Appl. 42 (1991), 95-99. | Zbl 0764.54023
[00004] [5] E. G. Effros, Transformation groups and C*-algebras, Ann. of Math. 81 (1965), 38-55. | Zbl 0152.33203
[00005] [6] F. B. Jones, The aposyndetic decomposition of homogeneous continua, Topology Proc. 8 (1983), 51-54. | Zbl 0537.54022
[00006] [7] J. Krasinkiewicz, On homeomorphisms of the Sierpiński curve, Comment. Math. Prace Mat. 12 (1969), 255-257. | Zbl 0235.54039
[00007] [8] P. Krupski, Recent results on homogeneous curves and ANR's, Topology Proc. 16 (1991), 109-118. | Zbl 0801.54015
[00008] [9] P. Krupski and J. R. Prajs, Outlet points and homogeneous continua, Trans. Amer. Math. Soc. 318 (1990), 123-141. | Zbl 0705.54026
[00009] [10] T. Maćkowiak, Terminal continua and the homogeneity, Fund. Math. 127 (1987), 177-186. | Zbl 0639.54027
[00010] [11] T. Maćkowiak and E. D. Tymchatyn, Continuous mappings on continua II, Dissertationes Math. (Rozprawy Mat.) 225 (1984). | Zbl 0584.54029
[00011] [12] J. C. Mayer, L. G. Oversteegen and E. D. Tymchatyn, The Menger curve. Characterization and extension of homeomorphisms of non-locally-separating closed subsets, ibid. 252 (1986). | Zbl 0649.54020
[00012] [13] P. Minc and J. T. Rogers, Jr., Some new examples of homogeneous curves, Topology Proc. 10 (1985), 347-356. | Zbl 0609.54027
[00013] [14] R. L. Moore, Triodic continua in the plane, Fund. Math. 13 (1929), 261-263. | Zbl 55.0978.03
[00014] [15] J. R. Prajs, Openly homogeneous continua having only arcs for proper subcontinua, Topology Appl. 31 (1989), 133-147. | Zbl 0668.54021
[00015] [16] J. T. Rogers, Jr., Decompositions of homogeneous continua, Pacific J. Math. 99 (1982), 137-144. | Zbl 0485.54028
[00016] [17] J. T. Rogers, An aposyndetic homogeneous curve that is not locally connected, Houston J. Math. 9 (1983), 433-440. | Zbl 0526.54019
[00017] [18] J. T. Rogers, Aposyndetic continua as bundle spaces, Trans. Amer. Math. Soc. 283 (1984), 49-55. | Zbl 0541.54040
[00018] [19] J. T. Rogers, Homogeneous curves that contain arcs, Topology Appl. 21 (1985), 95-101. | Zbl 0575.54031
[00019] [20] J. T. Rogers, Decompositions of continua over the hyperbolic plane, Trans. Amer. Math. Soc. 310 (1988), 277-291. | Zbl 0704.54020
[00020] [21] G. T. Whyburn, Analytic Topology, Amer. Math. Soc. Colloq. Publ. 28, Providence, R.I., 1942.
[00021] [22] G. T. Whyburn, Topological characterization of the Sierpiński curve, Fund. Math. 45 (1958), 320-324. | Zbl 0081.16904