Mauricio Achigar, Alfonso Artigue, Ignacio Monteverde
We study expansive dynamical systems from the viewpoint of general topology. We introduce the notions of orbit and refinement expansivity on topological spaces extending expansivity in the compact metric setting. Examples are given on non-Hausdorff compact spaces. Topological properties are studied in relation to separability axioms, metrizability and uniform expansivity.
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Expensive Homeomorphism of Convex Bodies
本文利用Borsuk-Ulam定理和Brouwer不动点定理，通过归纳与边界延拓方法，证明了欧几里得空间中凸体内不存在膨胀同胚，揭示了凸性对映射的拓扑约束。
On Stronger Forms of Expansivity
本研究提出正扩张映射的更强形式$p \mathscr{F}-$扩张映射的概念，构造相关实例、推导成立条件与性质，并研究半群下的$\mathscr{F}-$扩张同胚及逆极限空间移位映射的关联。
A Hausdorff topological group topology on a group $G$ is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on $G$. For every compact metrizable space $X$ containing an open $n$-cell, $n\ge2$, the homeomorphism group $H(X)$ has no minimum Hausdorff group topology. The homeomorphism groups of the Cantor set and the Hilbert cube have no minimum group topology. For every compact metrizable space $X$ containing a dense open one-manifold, $H(X)$ has the minimum group topology. Some, but not all, oligomorphic groups have the minimum group topology.
On Multi-Split Continuity and Split Homeomorphisms
该研究引入拓扑空间间的多分裂连续函数（弱连续性形式，推广分裂连续性且复合稳定），定义相关多函数，证明其与有限usco映射选择的联系，并引入分裂同胚，刻画紧正则豪斯多夫空间中带切割重粘合的变形。
We show that the homeomorphism group of a surface without boundary does not admit a Hausdorff group topology strictly coarser than the compact-open topology. In combination with known automatic continuity results, this implies that the compact-open topology is the unique Hausdorff separable group topology on the group if the surface is closed or the complement in a closed surface of either a finite set or the union of a finite set and a Cantor set.
Let X be a zero-dimensional compact space such that all non-empty clopen subsets of X are homeomorphic to each other, and let H(X) be the group of all self-homeomorphisms of X with the compact-open topology. We prove that the Roelcke compactification of H(X) can be identified with the semigroup of all closed relations on X whose domain and range are equal to X. We use this to prove that the group H(X) is topologically simple and minimal, in the sense that it does not admit a strictly coarser Hausdorff group topology.
Given uniformly homeomorphic metric spaces $X$ and $Y$, it is proved that the hyperspaces $C(X)$ and $C(Y)$ are uniformly homeomorphic, where $C(X)$ denotes the collection of all nonempty closed subsets of $X$, and is endowed with Hausdorff distance. Gerald Beer has proved that the hyperspace $C(X)$ is Atsuji when $X$ is either compact or uniformly discrete. An Atsuji space is a generalization of compact metric spaces as well as of uniformly discrete spaces. In this article, we investigate the space $C(X)$ when $X$ is Atsuji, and a class of Atsuji subspaces of $C(X)$ is obtained. Using the obt…
What does it mean for a shape to change continuously? Over the space of convex regions, there is only one "reasonable" answer. However, over a broader class of regions, such as the class of star-shaped regions, there can be many different "reasonable" definitions of continuous shape change. We consider the relation between topologies induced by a number of metrics over a number of limited categories of open bounded regions in n-dimensional Euclidean space. Specifically, we consider a homeomorphism-based metric; the Hausdorff metric; the dual-Hausdorff metric; the symmetric difference metric; a…
The Zariski Topology on Homeomorphism groups
本文研究了同胚群上的扎里斯基拓扑，证明了 Thompson 群 F 和 T 的扎里斯基拓扑与标准紧开拓扑一致，而 V 群的扎里斯基拓扑不可约，不满足 Hausdorff 或群拓扑条件。
Let $Δ$ be a foliation on a topological manifold $X$, $Y$ be the space of leaves, and $p: X \to Y$ be the natural projection. Endow $Y$ with the factor topology with respect to $p$. Then the group $\mathcal{H}(X, Δ)$ of foliated (i.e. mapping leaves onto leaves) homeomorphisms of $X$ naturally acts on the space of leaves $Y$, which gives a homomorphism $ψ: \mathcal{H}(X, Δ) \to \mathcal{H}(Y)$. We present sufficient conditions when $ψ$ is continuous with respect to the corresponding compact open topologies. In fact similar results hold not only for foliations but for a more general class of pa…
We investigate the mapping class groups of a class of non-Hausdorff topological spaces which includes finite spaces. We show that the mapping class group of a finite space is isomorphic to the homeomorphism group of its $T_0$ quotient. As a corollary, we show that every finite group is the mapping class group of some finite, $T_0$ space.
Let $G$ be a group of homeomorphisms of a topological space $X$. $G$ is $\textit{(properly) isometrizable}$ if there exists a $G$-invariant (proper) gauge structure on $X$. $G$ is $\textit{equiregular}$ if for every $x \in X$ and every open neighborhood $U$ of $x$ in $X$ there is an open neighborhood $V$ of $x$ in $X$ such that $cl(V) \subset U$ and every $y \in X$ has an open neighborhood $N_y$ with the property that for every $g \in G$, if $g(N_y) \cap cl(V) \neq \emptyset$, then $g(N_y) \subset U$. $G$ is $\textit{nearly proper}$ if for all compact subsets $A$ and $B$ of $X$, $cl$ ( $\bigcu…
A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. This paper mostly deals with the topological group $H_+(X)$ of order-preserving homeomorphisms of a compact linearly ordered connected space $X$. We provide a sufficient condition on $X$ under which the topological group $H_+(X)$ is minimal. This condition is satisfied, for example, by: the unit interval, the ordered square, the extended long line and the circle (endowed with its cyclic order). In fact, these groups are even $a$-minimal, meaning, in this setting, that the compact-o…
Group topologies on groups of bi-absolutely continuous homeomorphisms
本文研究双绝对连续同胚群上的群拓扑，证明 τ_ac 与 τ_co 间不存在中间可分群拓扑，并给出 Roelcke 预紧性结果。
We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of compact manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then $Y$ has a universal cover. We then show that, for $i$ sufficiently large, the fundamental group of $M_i$ has a surjective homeomorphism onto the group of deck transforms of $Y$. Finally, in the non-collapsed case where the $M_i$ have an additional uniform lower bound on volume, we prove that the kernels of these surjective maps are finite with a uniform bound on their cardinality. A number of theorems are also …
In the beginning of the 20th century, A. N. Whitehead and T. de Laguna proposed a new theory of space, known as {\em region-based theory of space}. They did not present their ideas in a detailed mathematical form. In 1997, P. Roeper has shown that the locally compact Hausdorff spaces correspond bijectively (up to homeomorphism and isomorphism) to some algebraical objects which represent correctly Whitehead's ideas of {\em region} and {\em contact relation}, generalizing in this way a previous analogous result of de Vries concerning compact Hausdorff spaces (note that even a duality for the cat…
About the space of continuous functions with open domain
本文提出了一种将具有开放域的连续函数空间转化为波兰空间的度量方法，并介绍了用于局部紧致、Hausdorff和可数空间同胚群的度量。
The homotopy types of directed path and trace spaces
构造饱和有向空间，使两点间迹空间同胚于正方形而有向路径空间有非平凡基本群，证明典范商映射非弱同伦等价，结论对正则路径亦成立。
We characterize all compact and Hausdorff spaces $X$ which satisfy that for every multiplicative bijection $ϕ$ on $C(X, I)$, there exist a homeomorphism $μ: X \to X$ and a continuous map $p: X \to (0, +\infty)$ such that $$ϕ(f) (x) = f(μ(x))^{p(x)}$$ for every $f \in C(X,I)$ and $x \in X$. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. {\bf 134} (2006), 1065-1075). Some related results on other semigroups of functions are also given.
One-dimensional definable topological spaces in o-minimal structures
该研究在o-极小结构框架下探究一维可定义拓扑空间的性质，通过分段分解与可定义同胚等方法给出多类拓扑的刻画，并应用于解决集合论拓扑相关公开猜想的可定义版本等问题。
CONTENTS: On Selective screenability and examples of R. Pol. Workshops and conferences: The Oxford Conference on Topology and Computer Science in Honour of Peter Collins and Mike Reed; Boise Extravaganza In Set Theory (BEST2006). Research announcements: The isometry group of the Urysohn space as a Levy group; Chasing Silver; Disjoint Non-Free Subgoups of Abelian Groups; Characterizing metric spaces whose hyperspaces are absolute neighborhood retracts; A Vitali set can be homeomorphic to its complement; o-Boundedness of free objects over a Tychonoff space; On the consistency strength of the Mil…
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