Funciones inducidas conexas

SERGIO A. PÉREZ^*
Universidad Industrial de Santander, Escuela Matemáticas, Bucaramanga, Colombia.

Resumen. Se dice que una función f : X → Y definida entre espacios topológicos es conexa si la gráfica Γ(f ) = {(x, f (x)) : x Є X} es conexa. Dado un continuo X, se consideran los hiperespacios: 2^X, la colección de todos los subconjuntos cerrados no vacíos de X; C(X), el conjunto de todos los subcontinuos de X; y F_n(X), los subconjuntos no vacíos de a lo más n puntos de X. Además, dada una función f : X → Y entre continuos, consideramos las funciones inducidas 2^f; : 2^X → 2^Y definidas por para cada A Є 2^X; F_n(f) : F_n(X) → Fn(Y), la función restricción F_n(f) = 2^f |F_n(X); y si f es una función de Darboux débil, definimos C(f) : C(X) → C(Y) por C(f) = 2^f |C(X). En este artículo estudiamos las relaciones entre las siguientes cinco afirmaciones: 1) f es conexa; 2) C(f) es conexa; 3) F_n(f) es conexa, para algún n ≥ 2; 4) F_n(f) es conexa, para todo n ≥ 2; 5) 2^f es conexa.
Palabras claves: Continuo, funciones inducidas, funciones conexas, funció Darboux débil, funciones casicontinuas.
MSC2010: 54E40, 54B20, 54C10.

Induced connected functions

Abstract. A function between topological spaces f : X → Y is said to be connected provided that the graph Γ(f) = {(x, f(x)) : x 2_X} is connected. Given a continuum X, some hyperspaces are considered: 2X, the collection of all non-empty closed subsets of X; C(X), the set of all subcontinua of X, and F_n(X) the set of nonempty subsets of at most n points of X. Moreover, given f : X → Y a function between continua, consider the induced functions: 2^f : 2^X → 2^Y , defined by for each A Є 2^X; F_n(f) : F_n(X) → F_n(Y), the restriction function F_n(f) = 2^f |F_n(X); and, if f is a weak Darboux function, we define C(f) : C(X) → C(Y) by C(f) = 2^f |C(X). In this paper we study the relationships between the following five statements: 1) f is connected; 2) C(f) is connected; 3) F_n(f ) is connected, for some n ≥ 2; 4) Fn(f) is connected, for all n ≥ 2; 5) 2^f is connected.
Keywords: Continuum, induced functions, connected functions, weak Darboux function, almost continuous functions.

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References

[1] Borsuk K. and Ulam S., "On symmetric products of topological spaces", Bull. Amer. Math. Soc. 37 (1931), no. 12, 875–882.

[2] Brown J.B., "Almost continuity of the Cesàro-Vietoris function", Proc. Amer. Math. Soc. 49 (1975), 185–188.

[3] Dugundji J., Topology, Allyn and Bacon, Inc., Boston, 1966.

[4] Garrett B.D., "When almost continuity implies connectivity", Topology Proc. 13 (1988), no. 2, 203–210.

[5] Illanes A., "Induced almost continuous functions on Hyperspaces", Colloq. Math. 105 (2006), no. 1, 69–76.

[6] Illanes A. and Nadler S.B. Jr., Hyperspaces. Fundamentals and Recent Advances, Monographs and textbooks in Pure and Applied Mathematics 216, Marcel Dekker, New York, 1999.

[7] Kellum K.R. and Rosen H., "Compositions of continuous functions and connected functions", Proc. Amer. Math. Soc. 115 (1992), no. 1, 145–149.

[8] Nadler S.B. Jr., "Continua on which all real-valued connected functions are connectivity functions", Topology Proc. 28 (2004), no. 1, 229–239.

[9] Nadler S.B. Jr., "Local connectivity functions on arcwise connected spaces and certain continua", Topology Appl. 153 (2006), no. 8, 1279–1290.

[10] Nadler S.B. Jr., Continuum Theory. An Introduction, Monographs and textbooks in Pure and Applied Mathematics, Vol. 158, Marcel Dekker, New York, 1992.

[11] Stallings J., "Fixed point theorems for connectivity maps", Fund. Math. 47 (1959), 249–263.

[12] Vietoris L., "Stetige Mengen", Monatsh. Math. Phys. 31 (1921), no. 1, 173–204.

^*E-mail :sergio.2060@hotmail.com.
Recibido: 24 de julio de 2013, Aceptado: 20 de agosto de 2013.