从曲面的补曲面识别曲面的不可分解子连续统

Approximations by disjoint continua and a positive entropy conjecture

E.D. Tymchatyn constructed a hereditarily locally connected continuum which can be approximated by a sequence of mutually disjoint arcs. We show the example re-opens a conjecture of G.T. Seidler and H. Kato about continua which admit positive entropy homeomorphisms. We prove that every indecomposable semicontinuum can be approximated by a sequence of disjoint subcontinua, and no composant of an indecomposable continuum can be embedded into a Suslinian continuum. We also prove that if $Y$ is a hereditarily unicoherent Suslinian continuum, then there exists $\varepsilon>0$ such that every two $\…

Singularities of meager composants and filament composants

Suppose $Y$ is a continuum, $x\in Y$, and $X$ is the union of all nowhere dense subcontinua of $Y$ containing $x$. Suppose further that there exists $y\in Y$ such that every connected subset of $X$ limiting to $y$ is dense in $X$. And, suppose $X$ is dense in $Y$. We prove $X$ is homeomorphic to a composant of an indecomposable continuum, even though $Y$ may be decomposable. An example establishing the latter was given by Christopher Mouron and Norberto Ordoñez in 2016. If $Y$ is chainable or, more generally, an inverse limit of identical topological graphs, then we show $Y$ is indecomposable …

不可度量化连续统超空间的维数

The dimension of hyperspaces of non-metrizable continua

证明了对任意Hausdorff连续统X，若dim X > 1，则X的子连续的超空间C（X）不是C-空间;若dim X = 1且X是遗传不可分解的，则dim C（X）= 2或C（X）不是C-空间。这推广了已知的度量连续统的结果。

The Shore Point Existence Problem is Equivalent to the Non-Block Point Existence Problem

We prove the three propositions are equivalent: $(a)$ Every Hausdorff continuum has two or more shore points. $(b)$ Every Hausdorff continuum has two or more non-block points. $(c)$ Every Hausdorff continuum is coastal at each point. Thus it is consistent that all three properties fail. We also give the following characterisation of shore points: The point $p$ of the continuum $X$ is a shore point if and only if there is a net of subcontinua in $\{K \in C(X): K \subset κ(p) - p\}$ tending to $X$ in the Vietoris topology. This contrasts with the standard characterisation which only demands the …

Diagonals separating the square of a continuum

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 t…