ABSTRACT
A metric, compact, connected space is a continuum. For a given positive integer n and a continuum X, denote by Fn(X] the space consisting of all nonempty subsets of X, having at most n elements, with the Vietoris topology (see 1.1 of [8] or 0.12 of [10]). This is the so called n
2. DEFINITIONS
By a graph we mean a continuum which can be written as the union of finitely many arcs any two of which are either disjoint, or intersect
only in one or both of their end points. By a segment of a graph X we shall always mean one of those arcs. The end points of the segments of X are called vertices of X. Given a point x e X and a natural number n, the order of X at re, denoted ord(x, X], is n provided that for every e > 0, there exists an open set U of X containing x with diam(U) < e such that dx(U) consists of exactly n points (see Theorem 9.12 of [11]). For each vertex v € X we have either ord(v, X) = I if v is an end point of X, or ord(v,X) > 2 otherwise. If ord(v,X) > 3, then v is called a ramification point of X. By a simple n-od (n > 3) we mean a graph X with only one ramification point, exactly n end points and without circles. A simple 3-od will be called simple triad. The complete graph Km is the graph with exactly m vertices such that any two vertices are joined by a segment of the graph. Let V be the set of vertices of a graph G, G is a bipartite graph with vertex clases V± and V-2 if V — Vi^V-2, Vi D V2 = 0 and each segment of G joins a vertex of V\ to a vertex of V^. The graph G is said bipartite complete if each vertex of Vi is joined to every vertex of V?, by segments of G, if | V\ j= m and | Vz \= n then G is denoted by Km,n. Given a continuum X, the cone over X is the decomposition space of the upper semicontinuous decomposition (X x [0, 1])/X x {!}. The cone over X will be denoted by cone(X). A space E is a link if E is homeomorphic with a disjoint union S^ U ... U 5^ of one or more circumferences in JR3. Given two circumferences a and j3 in ./R3, denote by lk(a, (3) the linking number of a and /5 (see pages 132-135 of [12]). A link E of two components Z/i, L2 is L-trivial if lk(Li, L2) = 0. Let X be a continuum. A nondegenerate subcontinuum AQ of X is called a convergence continuum (of X), or a continuum of convergence (of X), provided that there is a sequence {A} i^ of subcontinua Ai of X such that {A} 1^ converges to AQ and A,- n Aj — 0 for i, j e W U {0}. A figure eight continuum is a graph which is homeomorphic to the symbol representing the number eight. An n-noose is a graph which is homeomorphic to the union of one circle S1 and n arcs Ai,...,An such that there exists a point p 6 S1 with the property that for each i £ {!,..., n}, S1 D Ai = {p}, p is an end point of Ai for each i.
