N-SEQUENCES AND CONTRACTIBILITY IN HYPERSPACES JANUSZ J. CHARATONIK

Abstract. The existence of an N-sequence in a continuum is an obstruction that implies noncontractibility of the continuum. The aim of the present paper is to show that the existence of an N-sequence in the continuum X does not imply noncontractibility of some hyperspaces of X. 1. Preliminaries All considered spaces are assumed to be metric. We denote by N the set of all positive integers, and by R the space of reals. A continuum means a compact connected space, and a mapping means a continuous function. A curve means a one-dimensional continuum. A continuum is said to be uni-coherent provided that the intersection of any two of its subcontinua whose union is the whole continuum is connected. A continuum is said to be hereditarily uni-coherent provided that each of its subcontinua is unicoherent. If any two points of a space can be joined by an arc lying in the space, then the space is said to be arcwise connected. A dendroid means an arcwise connected and hereditarily unicoherent continuum. An end point of a dendroid X is defined as a point p of X which is an end point of each arc containing p. By a ramification point of a dendroid X we understand a point which is the center of a simple triod contained in X. A dendroid having at most one ramification point v is called a fan, and v is called its top. The cone over the Cantor middle-thirds set is called the Cantor fan. A continuum X is said to be uniformly arcwise connected provided that it is arcwise connected and that for each ε> 0 there is a k ∈ N such that every arc in X contains k points that cut it into subarcs of diameters less than ε. By [13