The Lelek fan is unique

Abstract. It is proven that a smooth fan with a dense set of end-points is unique. Some other characterizations of the fan are also given. A. Lelek has shown in [5, §9, page 314], an example of a fan with a one-dimensional set of its end-points. The fan is smooth and the set of its end-points is dense in it. In this paper it is proven that each such fan is homeomorphic to the Lelek example. As a consequence, each confluent image of the fan is homeomorphic to it. The only other continuum having this property, that is known to the author, is an arc (see [1, Corollary 20, page 32]), and there is a conjecture that the pseudo-arc is another one. Other characterizations of the Lelek fan are also obtained. Only metric spaces will be considered. A fan is an arcwise connected, hereditarily unicoherent continuum with at most one ramification-point (i.e., a point which is the unique common point of any two of three dis-tinct arcs). This point is called the top of the fan. An end-point of a fan is a point that is the end-point of each arc containing it. A fan with a top t is said to be smooth (see [1, §3, page 7]) if, for each sequence {Pn}~=l of its points converging to a point p, the arcs tPn (joining the top t with the point Pn) converge to the arc ip. Throughout this paper, the letter X always denotes a smooth fan with a dense set E ( X) of its end-points. We denote by C the Cantor ternary set in the unit interval, and by F the Cantor fan C x [0, l]/C x {OJ. We denote its points by (x, y) for x E C and y E [0, 1]. So, for each x E C, the point (x, 0) means the top of the fan F, and it is denoted by i. Given two sets A and B such that A C C and B C [0,1], the symbol A x B is always understood to be a respective subset of F. We denote by d a metric on F satisfying d ( (x, y) , (x, z)) = Iy- zl for all x E C and y, z E [0,1]. We will use a function 7r: F---+ [0,1] defined b