It is proved that the cylinder X × I over a planar λ-dendroid X has the fixed point property. This is a partial solution of two problems posed by R. H. Bing (cf. [1], Questions 9 and 10).

We give an example of an extremally disconnected compact Hausdorff space with an open continuous selfmap such that the fixed point set is nonvoid and nowhere dense, respṫhat there is exactly one nonisolated fixed point.

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...

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 $\mathbf{T}$.

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 simple topological measures X* on a q-space X are shown to be a superextension of X. Properties inherited from superextensions to topological measures are presented. The homology groups of various subsets of X* are calculated. For a q-space X, X* is shown to be a q-space. The homology of X* when X is the annulus is calculated. The homology of X* when X is a more general genus one space is investigated. In particular, X* for the torus is shown to have a retract homeomorphic to an infinite product...

An embedding X ⊂ G of a topological space X into a topological group G is called functorial if every homeomorphism of X extends to a continuous group homomorphism of G. It is shown that the interval [0, 1] admits no functorial embedding into a finite-dimensional or metrizable topological group.