Una introducción a los continuos homogéneos

SERGIO MACÍAS^*
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México D.F., C.P. 04510, MÉXICO.

Resumen. Un continuo es un espacio métrico, compacto y conexo. Un continuo X es homogéneo si para cualesquiera dos de sus puntos x₁ y x₂ de X, existe un homeomorfismo h: X X tal que h(x₁) = x₂. Presentaremos un poco de historia, ejemplos y propiedades de este tipo de continuos. Daremos una demostración del Teorema de descomposición aposindética de Jones.
Palabras claves: Círculo de pseudoarcos, continuo, cubo de Hilbert, curva universal de Menger, espacio homogéneo, función monótona, función T de Jones, pseudoarco.
MSC2000: 54C60, 54F20.

An introduction to homogeneous continua

Abstract. A continuum is a compact, connected, metric space. A continuum X is homogeneous provided that for each pair of points x₁ and x₂ of X, there exists a homeomorphism h: X X such that h(x₁) = x₂. We present a bit of history, examples and properties of this kind of continua. We give a proof of Jones's Aposyndetic Decomposition Theorem.
Keywords: Circle of pseudo-arcs, continuum, Hilbert cube, Menger universal curve, homogeneous space, monotone map, Jones's set function T , pseudoarc.

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Referencias

[1] Anderson R.D., "A characterization of the universal curve and a proof of its homogeneity", Ann. Math., 67 (1958), 313-324.         [ Links ]

[2] Anderson R.D., "One-dimensional continuous curves and a homogeneity theorem", Ann. Math., 68 (1958), 1-16.         [ Links ]

[3] Bellamy D.P. and Lum L., "The cyclic connectivity of homogeneous arcwise connected continua", Trans. Amer. Math. Soc., 266 (1981), 389-396.         [ Links ]

[4] Bing R.H., "A homogeneous indecomposable plane continuum", Duke Math. J., 15 (1948), 729-742.         [ Links ]

[5] Bing R.H., "Snake-like continua", Duke Math. J., 18 (1951), 653-663.         [ Links ]

[6] Bing R.H., "Concerning hereditarily indecomposable continua", Pacific J. Math., 1 (1951),43-51.         [ Links ]

[7] Bing R.H., "Each homogeneous nondegenerate chainable continuum is a pseudo-arc", Proc. Amer. Math. Soc., 10 (1959), 345-346.         [ Links ]

[8] Bing R.H., "A simple closed curve is the only homogeneous bounded homogeneous plane continuum that contains an arc", Canad. J. Math., 12 (1960), 209-230.         [ Links ]

[9] Bing R.H. and Jones F.B., "Another homogeneous plane continuum", Trans. Amer. Math. Soc., 90 (1959), 171-192.         [ Links ]

[10] Case J.H., "Another 1-dimensional homogeneous continuum which contains an arc", Pacific J. Math., 11 (1961), 455-469.         [ Links ]

[11] Charatonik J.J., "History of continuum theory", en Handbook of the history of general topology Vol. 2 (C. E. Aull y R. Lowen, editores), Kluwer Academic Publishers, Dordrecht, Boston, London (1998), 703-786.         [ Links ]

[12] Choquet G., "Prolongement d'homeomorphies. Ensembles topologiquement Charactérisation topologique individuelle des ensembles fermés totalement discontinus", C. R. Acad. Sci. Paris, 219 (1944), 542-544.         [ Links ]

[13] Cohen H.J., "Some results concerning homogeneous plane continua", Duke Math. J., 18 (1951), 467-474.         [ Links ]

[14] Dantzig D. van, "Ueber topologisch homogene kontinua", Fund. Math., 15 (1930), 102-125.         [ Links ]

[15] Effros E.G., "Transformation groups and C*-algebras", Ann. Math., (2) 81 (1965), 38-55.         [ Links ]

[16] Espinoza B. and Macías S., "On the set function R", Topology Appl., 154 (2007), 2988-2996.         [ Links ]

[17] Fearnley L., "The pseudo-circle is not homogeneous", Bull. Amer. Math. Soc., 75 (1969), 554-558.         [ Links ]

[18] Fernández L. and Macías S., "The set functions T and K and irreducible continua", Colloq. Math., 121 (2010), 79-91.         [ Links ]

[19] Hagopian C.L., "Homogeneous plane continua", Houston J. Math., 1 (1975), 35-41.         [ Links ]

20] Hagopian C.L., "A characterization of solenoids", Pacific J. Math., 68 (1977), 425-435.         [ Links ]

[21] Jones F.B., "Aposyndetic continua and certain boundary problems", Amer. J. Math., 63 (1941), 545-553.         [ Links ]

[22] Jones F.B., "Concerning non-aposyndetic continua", Amer. J. Math., 70 (1948), 403-413.         [ Links ]

[23] Jones F.B., "A note on homogeneous plane continua", Bull. Amer. Math. Soc., 55 (1949), 113-114.         [ Links ]

[24] Jones F.B., "On a certain type of homogeneous plane continuum", Proc. Amer. Math. Soc., 6 (1955), 735-740.         [ Links ]

[25] Jones F.B., "Use of a new technique in homogeneous continua", Houston J. Math., 1 (1975), 57-61.         [ Links ]

[26] Kennedy J. and Rogers J.T. Jr., "Orbits of the pseudocircle", Trans. Amer. Math. Soc., 296 (1986), 327-340.         [ Links ]

[27] Krasinkiewicz J., "On two theorems of Dyer", Colloq. Math., 50 (1986), 201-208.         [ Links ]

[28] Knaster B. and Kuratowski C., "Problèmes", Fund. Math., 1 (1920), 223.         [ Links ]

[29] Macías S., "Covering spaces of homogeneous continua", Topology Appl., 59 (1994), 157-177.         [ Links ]

[30] Macías S., "Un poco de continuos homogéneos", en Memorias del XXIX Congreso de Matemáticas de la Sociedad Matemática Mexicana. Aportaciones Matemáticas, Comunicaciones # 20, Sociedad Matemática Mexicana (1997), 109-116.         [ Links ]

[31] Macías S., Topics on continua, Pure and Applied Mathematics Series, Vol. 275, Chapman & Hall/CRC, Taylor & Francis Group, Boca Raton, London, New York, Singapore, 2005.         [ Links ]

[32] Macías S., "A class of one-dimensional, nonlocally connected continua for which the set function T is continuous", Houston J. Math., 32 (2006), 161-165.         [ Links ]

[33] Macías S., "Homogeneous continua for which the set function T is continuous", Topology Appl., 153 (2006), 3397-3401.         [ Links ]

[34] Macías S., "A decomposition theorem for a class of continua for which the set function T is continuous", Colloq. Math., 109 (2007), 163-170.         [ Links ]

[35] Macías S., "On the continuity of the set function K", Topology Proc., 34 (2009), 167-173.         [ Links ]

[36] Macías S., "On continuously irreducible continua", Topology Appl., 156 (2009), 2357-2363.         [ Links ]

[37] Macías S., "On the idempotency of the set function T ", Houston J. Math., 37 (2011), 1297-1305.         [ Links ]

[38] Macías S. and Nadler S.B. Jr., "On hereditarily decomposable homogeneous continua", Topology Proc., 34 (2009), 131-145.         [ Links ]

[39] Mazurkiewicz S., "Problème 14", Fund. Math., 2 (1921), 285.         [ Links ]

[40] Mazurkiewicz S., "Sur les continus homogènes", Fund. Math., 5 (1924), 137-146.         [ Links ]

[41] McCord M.C., "Inverse limit sequences with covering maps", Trans. Amer. Math. Soc., 114 (1965), 197-209.         [ Links ]

[42] Minc P. and Rogers J.T. Jr., "Some new examples of homogeneous curves", Topology Proc., 10 (1985), 347-356.         [ Links ]

[43] Moise E.E., "An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua", Trans. Amer. Math. Soc., 63 (1948), 581-594.         [ Links ]

[44] Moise E.E., "A note on the pseudo-arc", Trans. Amer. Math. Soc., 67 (1949), 57-58.         [ Links ]

[45] Nadler S.B. Jr., Continuum Theory, An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 158, Marcel Dekker, New York, Basel, Hong Kong, 1992.         [ Links ]

[46] Rogers J.T. Jr., "The pseudo-circle is not homogeneous", Trans. Amer. Math. Soc., 148 (1970), 417-428.         [ Links ]

[47] Rogers J.T. Jr., "An aposyndetic homogeneous curve that is not locally connected", Houston J. Math., 9 (1983), 433-440.         [ Links ]

[48] Rogers J.T. Jr., "Orbits of higher-dimensional hereditarily indecomposable continua", Proc. Amer. Math. Soc., 95 (1985), 483-486.         [ Links ]

[49] Rogers J.T. Jr., "Higher dimensional aposyndetic decompositions", Proc. Amer. Math. Soc., 131 (2003), 3285-3288.         [ Links ]

[50] Sierpinski W., "Sur une propiété topologique des ensembles dénombrables denses en soi", Fund. Math., 1 (1920), 11-16.         [ Links ]

[51] Waraszkiewicz Z., "Sur les courbes planes topologiquement homogènes", C. R. Acad. Sci. Paris, 204 (1937), 1388-1390.         [ Links ]

^*Autor para correspondencia: E-mail : sergiom@matem.unam.mx.
Recibido: 7 de septiembre de 2011, Aceptado: 7 de diciembre de 2011.