Abstract

PEREZ, SERGIO A. Induced connected functions. Integración - UIS [online]. 2013, vol.31, n.2, pp.121-132. ISSN 0120-419X.

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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