More on T-closed sets

Javier Camargo

https://orcid.org/0000-0003-1819-503X

Colombia

Industrial University of Santander

Escuela de Matemáticas, Facultad de Ciencias

Sergio Macías

https://orcid.org/0000-0001-7464-8852

Mexico

Universidad Nacional Autónoma de México

Instituto de Matemáticas

Submitted: 2025-03-01

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Accepted: 2025-10-07

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Published: 2026-01-22

DOI: https://doi.org/10.4995/agt.23481

Copyright (c) 2025 Sergio Macías, Javier Camargo

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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

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Keywords:

continuumwise connected space, continuum of colocal connectedness, hyperspace, property of Kelley, set functionT, T -closed set

Supporting agencies:

The first named author thanks the support given by "La Vicerrectoría de Investigación y Extensión de la Universidad Industrial de Santander y su Programa de Movilidad". The second named author thanks the Universidad Industrial de Santander, Colombia, for the support given during this research.

Abstract:

We consider properties of the diagonal of a continuum that are used later in the paper. We continue the study of T-closed subsets of a continuum X. We prove that for a continuum X, the statements: ΔX  is a nonblock subcontinuum of X2, ΔX is a shore subcontinuum of X2 and ΔX is not a strong centre of X2 are equivalent, this result answers in the negative Questions 35 and 36 and Question 38 ( i ∈ { 4,5 } ) of the paper "Diagonals on the edge of the square of a continuum, by A. Illanes, V. Martínez-de-la-Vega, J. M. Martínez-Montejano and D. Michalik". We also include an example, giving a negative answer to Question 1.2 of the paper"Concerning when F1 (X) is a continuum of colocal connectedness in hyperspaces and symmetric products, Colloquium Math., 160 (2020), 297-307'', by V. Martínez-de-la-Vega, J. M. Martínez-Montejano. We characterised the T-closed subcontinua of the square of the pseudo-arc. We prove that the T-closed sets of the product of two continua is compact if and only if such product is locally connected. We show that for a chainable continuum X, ΔX is a T-closed subcontinuum of X2 if and only if X is an arc. We prove that if X is a continuum with the property of Kelley, then the following are equivalent: ΔX is a T-closed subcontinuum of X2, X2\ΔX is strongly continuumwise connected, ΔX is a subcontinuum of colocal connectedness, and X2\ΔX is continuumwise connected. We give models for the families of T-closed sets and T-closed subcontinua of various families of continua.

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