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I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r_\sigma$, well-known in the theory of cardinal characteristics of the continuum. For a compact metrizable s …

This is a weak version of this problem, written down in Lviv Scottish Book. I start with necessary definitions. Let $K=-K$ be a centrally symmetric compact convex body in the Euclidean space $\mathbb …

I need a proper reference to the following obvious fact: An action of a group $G$ on a nonempty compact metrizable space $K$ is topologically transitive (= the orbit $GU$ of any nonempty open set …

It turns out that this problem is independent of ZFC because of the following simple Theorem. Under $\mathfrak t=\mathfrak c$, every topologically transitive continuous action of a group $G$ on $\ …

Let $G$ be a subgroup of the permutation group $S_\omega$ of the countable infinite set $\omega$. Each bijection $g\in G$ admits a unique extension to a homeomorphism $\bar g$ of the Stone-Cech compac …

By a dynamical system I understand a pair $(K,G)$ consisting a compact Hausdorff space and a subgroup $G$ of the homeomorphism group of $K$. We say that a dynamical system $(K,G)$ $\bullet$ is top …

The answer here is negative. Given any infinite-dimensional Banach space $X$, fix any non-zero linear continuous functional $f:X\to\mathbb R$ and fix any vector $x_1\in f^{-1}(1)$. Take any dense $G_ …

No solenoid can be embedded into the Mobius strip. To derive a contradiction, assume that some solenoid $S$ embeds into the Mobius strip $M$. Let $\pi:C\to M$ be a 2-fold covering map of the cylinder …

In this paper of J.Kwapisz I have found the following Theorem 1. Any homeomorphism $h$ of the dyadic solenoid $S$ is isotopic to the "affine" homeomorphism of the form $g:x\mapsto \pm(2^n x+b)$ for s …

The answer to both questions is affirmative. Theorem 1. The complete Erdos space $\mathfrak E_c$ has a self-homeomorphism whose every orbit is dense in $\mathfrak E_c$. Proof. We use a known result …