More on $\mathcal{T}$-closed sets

TL;DR

This paper investigates properties of the diagonal in continua, establishing equivalences among various $T$-closed set conditions, providing counterexamples, and characterizing $T$-closed subcontinua in specific continuum classes.

Contribution

It advances the understanding of $T$-closed sets in continua, proving key equivalences, characterizations, and providing counterexamples to open questions in continuum theory.

Findings

Equivalence of several properties of $ riangle_X$ in continua.

Counterexamples to previously posed questions.

Characterization of $T$-closed subcontinua in specific continua.

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: $\Delta_X$ is a nonblock subcontinuum of $X^2$, $\Delta_X$ is a shore subcontinuum of $X^2$ and $\Delta_X$ is not a strong centre of $X^2$ are equivalent, this result answers in the negative Questions 35 and 36 and Question 38 ($i\in\{4,5\}$) of the paper ``Diagonals on the edge of the square of a continuum, by A. Illanes, V. Mart\'inez-de-la-Vega, J. M. Mart\'inez-Montejano and D. Michalik''. We also include an example, giving a negative answer to Question 1.2 of the paper ``Concerning when $F_1(X)$ is a continuum of colocal connectedness in hyperspaces and symmetric products, Colloquium Math., 160 (2020), 297-307'', by V. Mart\'inez-de-la-Vega, J. M. Mart\'inez-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 $X$, $\Delta_X$ is a $T$-closed subcontinuum of $X^2$ 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: $\Delta_X$ is a $T$-closed subcontinuum of $X^2$, $X^2\setminus\Delta_X$ is strongly continuumwise connected, $\Delta_X$ is a subcontinuum of colocal connectedness, and $X^2\setminus\Delta_X$ is continuumwise connected. We give models for the families of $T$-closed sets and $T$-closed subcontinua of various families of continua.