Sobre los continuos tipo θ y θ_n

SERGIO MACÍAS^*

Universidad Nacional Autónoma de México, Instituto de Matemáticas, Circuito Exterior, Ciudad
Universitaria, México D.F., México.
University of Birmingham, School of Mathematics and Statistics, Birmingham, United Kingdom.

Resumen. En 1974 el Profesor R. W. FitzGerald definió los continuos tipo θ y θ_n. (Un continuo X es un continuo tipo θ (tipo θ_n, para algún número natural n) si para cada subcontinuo K de X, resulta que X \K sólo tiene un número finito de componentes (X \ K tiene a lo más n componentes).) Los profesores E. E. Grace y E. J. Vought continuaron el estudio de estas clases de continuos, cuando tales continuos admiten una descomposición monótona semicontinua superiormente cuyo cociente es una gráfica. El objetivo de este trabajo es presentar algunas de las propiedades de los continuos tipo θ y θ_n, principalmente cuando la descomposición mencionada anteriormente es continua [14].

Palabras clave: Continuo débilmente irreducible, continuo continuamente irreducible, continuo irreducible, continuo tipo θ, continuo tipo θ_n, descomposición continua, función T de Jones, hiperespacio, productos simétricos, subconjunto Z.
MSC2010: 54B20, 54C60.

On Type θ and θ_n continua

Abstract. In 1974 Professor R. W. FitzGerald defined type θ and θ_n continua. (A continuum X is of type θ (type θ_n, for some positive integer n) if for each subcontinuum K of X, we have that X \ K has only a finite number of components (X \ K has at most n components).) Professors E. E. Grace and E. J. Vought continued the study of these clases of continua, when such continua admit an upper semicontinuous monotone decomposition whose quotient space is a graph. The purpose of this work is to present some of the properties of type θ and θ_n continua, mainly when the decomposition is continuous [14].

Keywords: Continuous decomposition, continuously irreducible continuum, hyperspace, idempotency, Jones' set function T , irreducible continuum, symmetric products, type θ continuum, type θ_n continuum, type A′ θ-continuum, weakly irreducible continuum, Z-set.

Texto Completo disponible en PDF

Referencias

[1] Davis H.S."A note on connectedness im kleinen", Proc. Amer. Math. Soc. 19 (1968), 1237-1241.         [ Links ]

[2] Eberhart C. and Nadler S.B. Jr.,"Irreducible Whitney levels", Houston J. Math. 6 (1980), no. 3, 355-363.         [ Links ]

[3] Fernández L. and Macías S.,"The set functions T and K and irreducible continua", Colloq. Math. 121 (2010), no. 1, 79-91.         [ Links ]

[4] Fitzgerald R.W.,"Connected sets with a finite disconnection property", in Studies in Topology. Academic Press, (1975), 139-173.         [ Links ]

[5] Grace E.E.,"Monotone decompositions of θ-continua", Trans. Amer. Math. Soc. 275 (1983), no. 1, 287-295.         [ Links ]

[6] Grace E.E. and Vought E.J.,"Monotone decompositions of θ_n-continua", Trans. Amer. Math. Soc. 263 (1981), no. 1, 261-270.         [ Links ]

[7] Grace E.E. and Vought E.J.,"Quasimonotone mappings on θ_n-continua", Topology Appl. 17 (1984), no. 1, 55-62.         [ Links ]

[8] Grace E.E. and Vought E.J.,"Refinable maps and θ_n-continua", Proc. Amer. Math. Soc. 106 (1989), no. 1, 231-239.         [ Links ]

[9] Heath J.,"On n-ods", Houston J. Math. 9 (1983), no. 4, 477-487.         [ Links ]

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

[11] Macías S.,"Un Breve Panorama de los Hiperespacios de Continuos", Rev. Integr. Temas Mat. 23 (2005), no. 2, 1-13.         [ Links ]

[12] Macías S.,"A decomposition theorem for a class of continua for which the set function T is continuous", Colloq. Math. 109 (2007), no. 1, 163-170.         [ Links ]

[13] Macías S.,"On continuously irreducible continua", Topology Appl. 156 (2009), no. 14, 2357-2363.         [ Links ]

[14] Macías S.,"On continuously type A' θ_n-continua", manuscrito.         [ Links ]

[15] Macías S. and Nadler S.B. Jr.,"Z-sets in hyperspaces", Questions Answers Gen. Topology. 19 (2001), no. 2, 227-241.         [ Links ]

[16] Maćkowiak T.,"Continuous mappings on continua", Dissertationes Math. (Rozprawy Mat.) 158 (1979), 1-95.         [ Links ]

[17] Maćkowiak T.,"Singular arc-like continua", Dissertationes Math. (Rozprawy Mat.) 257 (1986), 1-35.         [ Links ]

[18] Mohler L. and Oversteegen L.G.,"On the structure of tranches in continuously irreducible continua", Colloq. Math. 54 (1987), no. 1, 23-28.         [ Links ]

[19] Nadler S.B. Jr., Hyperspaces of sets: A text with research questions, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, New York, Basel, 1978. Reprinted in: Aportaciones Matemáticas de la Sociedad Matemática Mexicana, Serie Textos # 33,         [ Links ] 2006.

[20] Thomas E.S. Jr.,"Monotone decompositons of irreducible continua", Rozprawy Mat. 50 (1966), 1-74.         [ Links ]

[21] Vought E.J.,"Monotone decompositions of continua not separated by any subcontinua", Trans. Amer. Math. Soc. 192 (1974), 67-78.         [ Links ]

[22] Vought E.J.,"Monotone decompositions of continua", in General Topology and Modern Analysis (ed. L. F. McAuley y M. M. Rao), Academic Press (1981), 105-113.         [ Links ]

[23] Ward L.E.,"Extending Whitney maps", Pacific J. Math. 93 (1981), no. 2, 465-469.         [ Links ]

^*Email: sergiom@matem.unam.mx; s.macias@bham.ac.uk
Recibido: 26 de septiembre de 2014, Aceptado: 20 de enero de 2015.
Para citar este artículo: S. Macías, Sobre los continuos tipo θ y θ_n, Rev. Integr. Temas Mat. 33 (2015), no. 1, 27-39.