The Lelek fan is unique. (English) Zbl 0675.54034

Charatonik, Włodzimierz J.

Houston J. Math. 15, No. 1, 27-34 (1989).

A fan is an arcwise connected hereditarily unicoherent continuum with at most one ramification point. A fan is smooth provided that for each sequence \([p_ n]\) of points converging to p, the arcs \(tp_ n\) converge to the arc tp, where t is the top of the fan. Such a continuum is embeddable in the Cantor fan. A. Lelek constructed a smooth fan with the property that the set of end-points is dense in the continuum, and the set of end-points together with the top is 1-dimensional connected set. The author proves that for a smooth fan different from an arc, the following are equivalent: (a) The fan Y is homeomorphic to the Lelek fan, (b) the set of end-points of Y is dense, (c) the set of end-points of Y together with the top is connected, (d) every confluent image of Y is homeomorphic to Y, (e) every monotone image of Y is homeomorphic to Y.

Reviewer: J.Grispolakis

MSC:

54F15

Continua and generalizations

Keywords:

fan; arcwise connected hereditarily unicoherent continuum; one ramification point; Cantor fan; smooth fan; Lelek fan; end-points; confluent image; monotone image