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 $\mathcal {T}(A)=\{x\in X : \text {for each subcontinuum }K\text { such that }x\in \textrm {Int}(K), \text { then }K\cap A\neq \emptyset \}.$ We show that $\mathcal {D}=\{\mathcal {T}(\{x\}):x\in X\}$ is a decomposition of $X$ when $\mathcal {T}$ is continuous (restricted to the hyperspace $2^{X}$). We present a characterization of the continuity of $\mathcal {T}$ and answer several open questions posed by D. Bellamy.