Let f be a map of a tree-like continuum M that sends each arc-component of M into itself. We prove that f has a fixed point. Hence every tree-like continuum has the fixed-point property for deformations (maps that are homotopic to the identity). This result answers a question of Bellamy. Our proof resembles an old argument of Brouwer involving uncountably many tangent curves. The curves used by Brouwer were originally defined by Peano. In place of these curves, we use rays that were originally defined...

In this paper we define a space σ(X) for approximate systems of compact spaces. The construction is due to H. Freudenthal for usual inverse sequences [4, p. 153–156]. We establish the following properties of this space: (1) The space σ(X) is a paracompact space, (2) Moreover, if X is an approximate sequence of compact (metric) spaces, then σ(X) is a compact (metric) space (Lemma 2.4). We give the following applications of the space σ(X): (3) If X is an approximate system of continua, then X = limX...

We consider the following topological spaces: $\mathbf{O}=\{z\in ℂ:|z+i|=1\}$, ${\mathbf{O}}_3=\mathbf{O}\cup \{z\in ℂ:z^4\in [0,1],Imz\ge 0\}$, ${\mathbf{O}}_4=\mathbf{O}\cup \{z\in ℂ:z^4\in [0,1]\}$, ${\infty }_1=\mathbf{O}:|z-i|=1\}\cup \{z\in ℂ:z\in [0,1]\}$, ${\infty }_2={\infty }_1\cup \{z\in ℂ:z^2\in [0,1]\}$, et $\mathbf{T}=\{z\in ℂ:z=\cos (3\theta )e^{i\theta },0\le \theta \le 2\pi \}$. Set $E\in \{{\mathbf{O}}_3,{\mathbf{O}}_4,{\infty }_1,{\infty }_2,\mathbf{T}\}$. An $E$ map $f$ is a continuous self-map of $E$ having the branching point fixed. We denote by $Per\left(f\right)$ the set of periods of all periodic points of $f$. The set $K\subset \mathbb{N}$ is the full periodicity kernel of $E$ if it satisfies the following two conditions: (1) If $f$ is an $E$ map and $K\subset Per\left(f\right)$, then $Per\left(f\right)=\mathbb{N}$. (2) If $S\subset \mathbb{N}$ is a set such that for every $E$ map $f$, $S\subset Per\left(f\right)$ implies $Per\left(f\right)=\mathbb{N}$, then $K\subset S$. In this paper we compute the full periodicity kernel of ${\mathbf{O}}_3,{\mathbf{O}}_4,{\infty }_1,{\infty }_2$ and $\mathbf{T}$.

We disprove the existence of a universal object in several classes of spaces including the class of weakly Lindelöf Banach spaces.

In this paper, we study the relation between a fuzzy measure and a fuzzy metric which is induced by the fuzzy measure. We also discuss some basic properties of the constructed fuzzy metric space. In particular, we show that the nonatom of fuzzy measure space can be characterized in the constructed fuzzy metric space.

We establish a relation between covering properties (e.gĿindelöf degree) of two standard topological spaces (Lemmas 4 and 5). Some cardinal inequalities follow as easy corollaries.

The aim of this paper is to prove the generalized Schoenflies theorem for the class of absolute suspensions. The question whether the finite-dimensional absolute suspensions are homeomorphic to spheres remains open. Partial solution to this question was obtained in [Sz] and [Mi]. Morton Brown gave in [Br] an ingenious proof of the generalized Schoenflies theorem. Careful analysis of his proof reveals that modulo some technical adjustments a similar argument gives an analogous result for the class...

It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving homeomorphism is moreover conjugate to all of its powers. It follows that the generic measure preserving homeomorphism extends to an action of (ℚ, +) by measure preserving homeomorphisms, and, in fact, to an action of the locally compact ring 𝔄 of finite adèles. ...

A lamination is a continuum which locally is the product of a Cantor set and an arc. We investigate the topological structure and embedding properties of laminations. We prove that a nondegenerate lamination cannot be tree-like and that a planar lamination has at least four complementary domains. Furthermore, a lamination in the plane can be obtained by a lakes of Wada construction.

The Golomb space ${\mathbb{N}}_{\tau }$ is the set $\mathbb{N}$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space ${\mathbb{N}}_{\tau }$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.

We examine the Gruenhage property, property * (introduced by Orihuela, Smith, and Troyanski), fragmentability, and the existence of σ-isolated networks in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO-spaces), and monotonically normal spaces. We show that any monotonically normal space with property * or with a σ-isolated network must be hereditarily paracompact, so that property * and the Gruenhage property are equivalent in monotonically normal spaces....