我们首先研究了连续统对角线的性质，这些性质将在论文后续部分使用。我们继续开展连续统$X$的$T$-闭子集的相关研究。我们证明，对于连续统$X$，以下三个命题是等价的：$Δ_X$是$X^2$的非阻塞子连续统、$Δ_X$是$X^2$的支撑子连续统、$Δ_X$不是$X^2$的强中心，该结果否定回答了A. Illanes、V. Martínez-de-la-Vega、J. M. Martínez-Montejano和D. Michalik发表的《连续统平方边缘的对角线》一文中的问题35、问题36以及问题38（$i\in\{4,5\}$）。我们还给出了一个例子，否定回答了V. Martínez-de-la-Vega与J. M. Martínez-Montejano发表在《Colloquium Math.》2020年第160卷297-307页的《关于$F_1(X)$何时为超空间与对称乘积中的局部余连通连续统》一文中的问题1.2。我们刻画了伪弧平方的$T$-闭子连续统的特征。我们证明，两个连续统的乘积的$T$-闭集族是紧的，当且仅当该乘积是局部连通的。我们表明，对于可链连续统$X$，$Δ_X$是$X^2$的$T$-闭子连续统当且仅当$X$是弧。我们证明，若$X$是具有Kelley性质的连续统，则以下命题等价：$Δ_X$是$X^2$的$T$-闭子连续统、$X^2\setminusΔ_X$是强连续统连通的、$Δ_X$是局部余连通子连续统、$X^2\setminusΔ_X$是连续统连通的。我们还给出了各类连续统族的$T$-闭集族与$T$-闭子连续统族的模型。
英文摘要：
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: $Δ_X$ is a nonblock subcontinuum of $X^2$, $Δ_X$ is a shore subcontinuum of $X^2$ and $Δ_X$ 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. 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 $F_1(X)$ 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 $X$, $Δ_X$ 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: $Δ_X$ is a $T$-closed subcontinuum of $X^2$, $X^2\setminusΔ_X$ is strongly continuumwise connected, $Δ_X$ is a subcontinuum of colocal connectedness, and $X^2\setminusΔ_X$ is continuumwise connected. We give models for the families of $T$-closed sets and $T$-closed subcontinua of various families of continua.
Characterizing $ (\mathcal{F}, \mathcal{G}) $-syndetic, $ (\mathcal{F}, \mathcal{G}) $-thick, and related notions of size using derived sets along ultrafilters
Connectedness of self-affine sets with product digit sets
设$T(A,\mathcal{D})$是由扩展矩阵$A=\left[\begin{array}{rr} p & 0\cr -a & q \end{array}\right]$和乘积数字集$\mathcal{D}=\{0,1,\dots,m-1\}\times \{0,1,\dots,n-1\}$生成的自仿射集.给出了$T(A,\mathcal{D})$连通的一个充要条件，推广了已有的结果.
