Shore and Non-Block Points in Hausdorff Continua

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 …

割点与非割点的Whitney性质及Whitney可逆性质

Whitney Properties and Whitney reversible properties of Cut and Non-Cut Points

本文研究割点与非割点的Whitney性质，证明岸点性质在可精化映射下保持，并指出块点、非岸点及强中心点非Whitney可逆，相关连续统类无公共模型。

连续统中闭集的补集的连通度

Connectivity degrees of complements of closed sets in continua

本文扩展了连续统中补集连通点与贫集的研究，引入闭集补集的连通度，并证明在 $n=1$ 且 $A$ 为贫集或单点集时新定义等价于已知非割点/集定义。

Metric properties of mean wiggly continua

We study lower and upper bounds of the Hausdorff dimension for sets which are wiggly at scales of positive density. The main technical ingredient is a construction, for every continuum K, of a Borel probabilistic measure μwith the property that on every ball B(x,r), with x in K, the measure is bounded by a universal constant multiple of r\exp(-g(x,r)), where g(x,r) > 0 is an explicit function. The continuum K is mean wiggly at exactly those points x in K where g(x, r) has a logarithmic growth to infinity as r goes to 0. The theory of mean wiggly continua leads, via the product formula for dime…

不可嵌入连续统的真类

Proper classes of non-embeddable continua

本文构造了一类道路连通、局部连通的紧致豪斯多夫空间，其中不同空间之间不存在连续单射，并给出了该类中空间的数量、权重和大小，以及可度量化情形下的平面嵌入性质。

The dimension of hyperspaces of non-metrizable continua

We prove that, for any Hausdorff continuum X, if dim X > 1 then the hyperspace C(X) of subcontinua of X is not a C-space; if dim X = 1 and X is hereditarily indecomposable then dim C(X) = 2 or C(X) is not a C-space. This generalizes results known for metric continua.