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

Abstract

Given a continuum $X$, for each $A\subseteq X$, the Jones' set function $\mathcal{T}$ is defined by $\mathcal{T}(A)=\{x\in X:\text{for each subcontinuum }K\text{ such that }x\in \text{Int}(K),\text{ then }K\cap A\ne \varnothing \}.$ 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.