Induced mappings between quotient spaces of n-fold hyperspaces of continua

Abstract

Abstract. For a continuum X the hyperspace of nonempty closed subsets of X with at most n components is called the n-fold hyperspace Cn(X) and if m < n then Cm(X) ⊂ Cn(X) so it is possible to form a quotient space Cn(X)/Cm(X) identifying the set Cm(X) to a point in Cn(X). If f is a mapping from a continuum X onto a continuum Y there will be a induced mappings between Cn(X) and Cm(X) and between the quotient spaces Cn(X)/Cm(X) and Cn(Y )/Cm(Y ). Now if a list of function properties that are of interest to continua theorists is considered, there will be natural questions about when these properties are passed on from the functions between the continua to the induced mappings between the hyperspaces or the induced mappings between the quotients of the hyperspaces. Many of these questions have been considered extensively for the hyperspaces so the main thing that is new here is the questions and answers about the quotient spaces and their induced mappings. Here we consider the following families of mappings:

Induced mappings on $C_n(X)/{C_n}_K(X)$

“…Readers especially interested in this topic are referred, for example, to [5], [7], [8], [11], [12], [16], [17]. Regarding induced mappings in quotient hyperspaces we refer the reader, for example, to [1], [3], [4], [6], [10], [30].…”

“…Example 2. In the interval I, we identify the point 0 with the point 1 3 , and the point 2 3 with the point 1. Let g be the quotient mapping, note that g is onto and is not weakly confluent.…”

“…Theorem 3. Let f : X → Y be a mapping between continua and n ∈ N. Then the following conditions are equivalent: (1)…”

“…Theorem 4. Let f : X → Y be a mapping between continua and n ∈ N. Consider the following conditions: (1) 1) ⇒ (4), ( 2) ⇒ (4), and (3) ⇒ (4).…”