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

TL;DR

This paper investigates the continuity properties of Jones' set function on continua, characterizes when it is continuous, and explores the resulting decompositions of the space, addressing open questions in the field.

Contribution

It provides a characterization of the continuity of Jones' set function and describes the resulting decompositions of the continuum, advancing understanding of its topological properties.

Findings

$ ext{D}=ig{\

ext{Decomposition of }X ext{ when } ext{T} ext{ is continuous}

Abstract

Given a continuum $X$, for each $A\subseteq X$, the Jones' set function $\mathcal{T}$ is defined by $\mathcal{T}(A)=\{x\in X : \text{for each subcontinuum }K\text{ such that }x\in \textrm{Int}(K), \text{ then }K\cap A\neq\emptyset\}.$ We show that $\mathcal{D}=\{\mathcal{T}(\{x\}):x\in X\}$ is decomposition of $X$ when $\mathcal{T}$ is continuous. We present a characterization of the continuity of $\mathcal{T}$ and answer several open questions posed by D. Bellamy.