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


Authors: Javier Camargo and Carlos Uzcátegui
Journal: Proc. Amer. Math. Soc. 145 (2017), 893-899
MSC (2010): Primary 54B20; Secondary 54C60
DOI: https://doi.org/10.1090/proc/13379
Published electronically: October 3, 2016
MathSciNet review: 3577889
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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.


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Javier Camargo
Affiliation: Escuela de Matemáticas, Facultad de Ciencias, Universidad Industrial de Santander, Ciudad Universitaria, Carrera 27 Calle 9, Bucaramanga, Santander, A.A. 678, Colombia
Email: jcamargo@saber.uis.edu.co

Carlos Uzcátegui
Affiliation: Escuela de Matemáticas, Facultad de Ciencias, Universidad Industrial de Santander, Ciudad Universitaria, Carrera 27 Calle 9, Bucaramanga, Santander, A.A. 678, Colombia – and – Centro Interdisciplinario de Lógica y Álgebra, Facultad de Ciencias, Universidad de Los Andes, Mérida, Venezuela
Email: cuzcatea@saber.uis.edu.co

DOI: https://doi.org/10.1090/proc/13379
Keywords: Jones' decomposition theorem, set function $\mathcal{T}$, continuum.
Received by editor(s): December 14, 2015
Received by editor(s) in revised form: April 15, 2016
Published electronically: October 3, 2016
Additional Notes: The authors thank La Vicerrectoría de Investigación y Extensión de la Universidad Industrial de Santander for the financial support for this work, which is part of the VIE project #1873.
Communicated by: Michael Wolf
Article copyright: © Copyright 2016 American Mathematical Society