Topics on Continua

Continuity of the Jones’ set function $\mathcal {T}$

“…If D = {T ({x}) : x ∈ X} is a decomposition of X, then it is easy to see that p / ∈ T ({q}) if and only if q / ∈ T ({p}), for p, q ∈ X, p = q; i.e., X is point T −symmetric. Thus, by Theorem 3.7 and [14, Corollary 3.2.15], we have the following result which give us a positive answer to Question 7.2.1 of [14].…”

“…In [14,Theorem 3.2.8], it is proved that if T is continuous then T is idempotent. The following result characterizes the continuity of T in terms of idempotence.…”

“…The set function T was defined by F. Burton Jones [7] in order to study some properties related with aposyndesis of continua. Since then it has been studied extensively as an aid to classify continua (see for instance [1], [2], [3], [4], [7], [8], [12], [10], [11], [13] and [14]).…”

On generic and dense chaos for maps induced on hyperspaces

“…We will give here only those we will need in the following text. For the proofs and further study of topology of hyperspaces see, e.g., [4], [5] and [6].…”

A 2-equivalent Kelley continuum

“…Let X and Y be continua, a map f : X → Y is said to be confluent provided that for any subcontinuum B of Y and any component A of f −1 (B), f (A) = B. A monotone map f : X → Y , is a map such that f −1 (C) is a connected set, for every connected subset C of Y (see [9,Lema 2.1.12,p. 74]).…”