More on T-closed sets

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.