Revista Integración, temas de matemáticas.

Revista Integración, temas de matemáticas

Vol. 31 No. 2 (2013): Revista Integración, temas de matemáticas
Research and Innovation Articles

Induced connected functions

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  • Sergio A. Pérez
Sergio A. Pérez
Universidad Industrial de Santander

Published 2013-12-17

Keywords

  • Continuum,
  • induced functions,
  • connected functions,
  • weak Darboux function,
  • almost continuous functions

How to Cite

Pérez, S. A. (2013). Induced connected functions. Revista Integración, Temas De matemáticas, 31(2), 121–132. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/3751

Abstract

A function between topological spaces f : X → Y is said to be connected provided that the graph Γ(f) = {(x, f(x)) : x ∈ 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 Fn(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: 2f: 2X → 2Y defined by 2f(A) = f(A) for each A ∈  2X; Fn(f): Fn(X) → Fn(Y),  the restriction function Fn(f) = 2f|Fn(X); and, if f is a weak Darboux function, we define C(f): C(X) → C(Y) by C(f) = 2f|C(X). In this paper we study the relationships between the following five statements: 1) f is connected; 2) C(f) is connected; 3) Fn(f) is connected, for some n ≥ 2; 4) Fn(f) is connected, for all n ≥ 2; 5) 2f is connected.

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