Complexity of atriodic continua

A property equivalent to being semi-Kelley

We present a property equivalent to the property of being semi-Kelley. Using this equivalence we prove that being semi-Kelley is a hereditary property for atriodic continua. We prove that semi-Kelley remainders are atriodic, moreover, we prove that semi-Kelley continua are semi-Kelley remainders for chainable continua, circularly chainable continua, and arc continua, and we give an example of an atriodic Kelley continuum which is a semi-Kelley remainder and not a Kelley remainder. We also prove that hereditarily semi-Kelley dendroids are smooth.

On the descriptive complexity of homogeneous continua

It is shown that the family of all homogeneous continua in the hyperspace of all subcontinua of any finite-dimensional Euclidean cube or the Hilbert cube is an analytic subspace of the hyperspace which contains a topological copy of the linear space $c_0=\{(x_k)\in \mathbb R^ω: \lim x_k=0\}$ as a closed subset.

The Complexity of the Classification Problems of Finite-Dimensional Continua

We consider the homeomorphic classification of finite-dimensional continua as well as several related equivalence relations. We show that, when $n \geq 2$, the classification problem of $n$-dimensional continua is strictly more complex than the isomorphism problem of countable graphs. We also obtain results that compare the relative complexity of various equivalence relations.

The complexity of the classification problem of continua

We prove that the homeomorphism problem for connected compact metric spaces is Borel bireducible with a universal orbit equivalence relation induced by a Borel action of a Polish group.

链状连续统上的线性序

Linear orders on chainable continua

研究链状连续统上依赖于链序列和超滤子的线性序，证明弧上有2种、Warsaw正弦曲线有4种、Knaster连续统有2^c种不同超滤子序，并刻画Suslinean性质与序同构于[0,1]的关系。

波兰空间上连通性关系的复杂性

The Complexity of Connectedness Relations on Polish Spaces

本文研究了波兰空间上三种连通性关系的复杂性，发现弧连通关系具有低复杂性，而连续统连通关系则具有更高复杂性。