{
  "id": "2407.09258",
  "version": "v1",
  "published": "2024-07-12T13:40:23.000Z",
  "updated": "2024-07-12T13:40:23.000Z",
  "title": "More on $\\mathcal{T}$-closed sets",
  "authors": [
    "Javier Camargo",
    "Sergio Macías"
  ],
  "categories": [
    "math.GN"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-12T13:40:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54B20",
      "54C60",
      "54F15"
    ],
    "keywords": [
      "closed sets",
      "closed subcontinuum",
      "colocal connectedness",
      "symmetric products",
      "nonblock subcontinuum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}