Topologically subordered rectifiable spaces and compactifications

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$ we have $\phi (x, x)=(x, e)$, where $\pi_{1}: G\times G\rightarrow G$ is the projection to the first coordinate. In this paper, we mainly discuss the rectifiable spaces which are suborderable, and show that if a rectifiable space is suborderable then it is metrizable or a totally disconnected P-space, which improves a theorem of A.V. Arhangel'ski\v\i\ in \cite{A20092}. As an applications, we discuss the remainders of the Hausdorff compactifications of GO-spaces which are rectifiable, and we mainly concerned with the following statement, and under what condition $\Phi$ it is true. Statement: Suppose that $G$ is a non-locally compact GO-space which is rectifiable, and that $Y=bG\setminus G$ has (locally) a property-$\Phi$. Then $G$ and $bG$ are separable and metrizable. Moreover, we also consieder some related matters about the remainders of the Hausdorff compactifications of rectifiable spaces.Comment: 14 pages (replace

Reconstructing Compact Metrizable Spaces

The deck, $\mathcal{D}(X)$, of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}]\colon x \in X\}$, where $[Y]$ denotes the homeomorphism class of $Y$. A space $X$ is (topologically) reconstructible if whenever $\mathcal{D}(Z)=\mathcal{D}(X)$ then $Z$ is homeomorphic to $X$. It is known that every (metrizable) continuum is reconstructible, whereas the Cantor set is non-reconstructible. The main result of this paper characterises the non-reconstructible compact metrizable spaces as precisely those where for each point $x$ there is a sequence $\langle B_n^x \colon n \in \mathbb{N}\rangle$ of pairwise disjoint clopen subsets converging to $x$ such that $B_n^x$ and $B_n^y$ are homeomorphic for each $n$, and all $x$ and $y$. In a non-reconstructible compact metrizable space the set of $1$-point components forms a dense $G_\delta$. For $h$-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense $G_\delta$ set of $1$-point components are presented, some reconstructible and others not reconstructible.Comment: 15 pages, 2 figure

Notes on the Schreier graphs of the Grigorchuk group

The paper is concerned with the space of the marked Schreier graphs of the Grigorchuk group and the action of the group on this space. In particular, we describe an invariant set of the Schreier graphs corresponding to the action on the boundary of the binary rooted tree and dynamics of the group action restricted to this invariant set.Comment: 33 pages, 4 figure

From continua to R-trees

We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X.Comment: This is the version published by Algebraic & Geometric Topology on 1 November 200