It is proved that any two composants of any two solenoids are homeomorphic.

The Cantor set and the set of irrational numbers are examples of 0-dimensional, totally disconnected, homogeneous spaces which admit elegant characterizations and which play a crucial role in analysis and dynamical systems. In this paper we will start the study of 1-dimensional, totally disconnected, homogeneous spaces. We will provide a characterization of such spaces and use it to show that many examples of such spaces which exist in the literature in various fields are all homeomorphic. In particular,...

We show that the Bruschlinsky group with the winding order is a homomorphism invariant for a class of one-dimensional inverse limit spaces. In particular we show that if a presentation of an inverse limit space satisfies the Simplicity Condition, then the Bruschlinsky group with the winding order of the inverse limit space is a dimension group and is a quotient of the dimension group with the standard order of the adjacency matrices associated with the presentation.

For any ordinal $\lambda$ of uncountable cofinality, a $\lambda$-tree is a tree $T$ of height $\lambda$ such that $|T_{\alpha }|<\mathrm{cf}\left(\lambda \right)$ for each $\alpha <\lambda$, where $T_{\alpha }=\{x\in T:\mathrm{ht}\left(x\right)=\alpha \}$. In this note we get a Pressing Down Lemma for $\lambda$-trees and discuss some of its applications. We show that if $\eta$ is an uncountable ordinal and $T$ is a Hausdorff tree of height $\eta$ such that $|T_{\alpha }|\le \omega$ for each $\alpha <\eta$, then the tree $T$ is collectionwise Hausdorff if and only if for each antichain $C\subset T$ and for each limit ordinal $\alpha \le \eta$ with $\mathrm{cf}\left(\alpha \right)>\omega$, $\left\{\mathrm{ht}\right(c):c\in C\}\cap \alpha$ is not stationary in $\alpha$. In the last part of this note, we investigate some...

In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if $g:Y\to X$ is a quasi-homeomorphism, $Z$ a sober space and $f:Y\to Z$ a continuous map, then there exists a unique continuous map $F:X\to Z$ such that $F\circ g=f$. Let $X$ be a $T_0$-space, $q:X\to {}^{s}X$ the injection of $X$ onto its sobrification ${}^{s}X$. It is shown, here, that $q\left(\text{Gold}\left(X\right)\right)=\text{Gold}\left({}^{s}X\right)$, where $\text{Gold}\left(X\right)$ is the set of all locally closed points of $X$. Some applications are also indicated. The Jacobson prime spectrum...

Let G ⊂ Homeo(E) be a group of homeomorphisms of a topological space E. The class of an orbit O of G is the union of all orbits having the same closure as O. Let E/G̃ be the space of classes of orbits, called the quasi-orbit space. We show that every second countable T₀-space Y is a quasi-orbit space E/G̃, where E is a second countable metric space. The regular part X₀ of a T₀-space X is the union of open subsets homeomorphic to ℝ or to 𝕊¹. We give a characterization of the spaces X with finite...

Let $C(X,\mathbb{Z})$, $C(X,\mathbb{Q})$ and $C\left(X\right)$ denote the $\ell$-groups of integer-valued, rational-valued and real-valued continuous functions on a topological space $X$, respectively. Characterizations are given for the extensions $C(X,\mathbb{Z})\le C(X,\mathbb{Q})\le C\left(X\right)$ to be rigid, major, and dense.