Resumen de Retractions from C(X) onto X and continua of type N

Félix Capulín Pérez, Wlodzimier J. Charatonik

• If a metric continuum X is of type N, then there is no a retraction from the hyperspace of subcontinua C(X) onto F1(X), and X admits no mean. We also give an example which answers a question posed by T. J. Lee. The question is the following: Is there a fan X without the bend intersection property such that X is not of type N? The answer is affirmative, we show a fan.

  Let A be a closed subset of X. A retraction is a mapping r from X onto A such that r restricted to A is the identity in A . A mean is a mapping m from X× X onto X such that a) m((x,x))=x for each x in X, b) m((x,y))=m((y,x)) for each x, y in X.

  X is a continuum of Type N if there exist in X an arc A=[p,q], two sequences of arcs An=[pn,p'n] and Bn=[qn,q'n] and points p"n in Bn--{qn,q 'n} and q"n in An--{pn,p'n} such that: 1. the sequences of arcs {An} and {Bn} converge to the arc A; 2. the sequences {pn}, {p'n} and {p"n} converge to the point p 3. the sequences {qn}, {q'n} and {q"n} converge to the point q; 4. each arc in X joining pn and p'n contains q"n; 5. each arc in X joining qn and q'n contains p"n.