对角线分隔连续统的平方
Diagonals separating the square of a continuum
摘要：
一个度量连续统$X$是不可分解的，如果它不能作为它的两个真子连续统的并集。$X$的子集$R$被称为连续连通的，条件是对于每对点$p,q\in R$，存在$X$的子连续统$M$，使得$\{p,q\}\subset M\subset R$。令$X^{2}$表示$X$的笛卡尔平方，$Δ$表示$X^{2}$的对角线。在\cite{ka}中，它被问到对于不同于弧的连续$X$，$X^{2}\setminus Δ$是连续连通的当且仅当$X$是可分解的。在本文中，我们表明，在这个问题中没有任何含义。为了证明不必要性，我们利用康托尔集到自身上的一个合适的同胚的动力学性质构造了一个合适的不可分解连续统$X$ .
英文摘要：
A metric continuum $X$ is indecomposable if it cannot be put as the union of two of its proper subcontinua. A subset $R$ of $X$ is said to be continuumwise connected provided that for each pair of points $p,q\in R$, there exists a subcontinuum $M$ of $X$ such that $\{p,q\}\subset M\subset R$. Let $X^{2}$ denote the Cartesian square of $X$ and $Δ$ the diagonal of $X^{2}$. In \cite{ka} it was asked if for a continuum $X$, distinct from the arc, $X^{2}\setminus Δ$ is continuumwise connected if and only if $X$ is decomposable. In this paper we show that no implication in this question holds. For the proof of the non-necessity, we use the dynamic properties of a suitable homeomorphism of the Cantor set onto itself to construct an appropriate indecomposable continuum $X$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
