arXiv subjects

Slawomir Turek

Publications and source records attributed to Slawomir Turek.

4 recordsLinked to original sources

arXiv2018

On continuous self-maps and homeomorphisms of the Golomb space

The Golomb space $\mathbb N_τ$ is the set $\mathbb N$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions $\{a+bn\}_{n=0}^\infty$ with coprime $a,b$. We prove that the Golomb space $\mathbb N_τ$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $Π$ of prime numbers is a dense metrizable subspace of $\mathbb N_τ$, and each homeomorphism $h$ of $\mathbb N_τ$ has the following properties: $h(1)=1$, $h(Π)=Π$ and $Π_{h(x)}=h(Π_x)$ for all $x\in\mathbb N$. Here by $Π_x$ we denote the set of prime divisors of $x$.

math.GN↗

arXiv2013

The groups $S^3$ and $SO(3)$ have no invariant binary $k$-network

A family $\mathcal N$ of closed subsets of a topological space $X$ is called a {\em closed $k$-network} if for each open set $U\subset X$ and a compact subset $K\subset U$ there is a finite subfamily $\mathcal F\subset\mathcal N$ with $K\subset\bigcup\F\subset \mathcal N$. A compact space $X$ is called {\em supercompact} if it admits a closed $k$-network $\mathcal N$ which is {\em binary} in the sense that each linked subfamily $\mathcal L\subset\mathcal N$ is centered. A closed $k$-network $\mathcal N$ in a topological group $G$ is {\em invariant} if $xAy\in\mathcal N$ for each $A\in\mathcal N$ and $x,y\in G$. According to a result of Kubiś and Turek, each compact (abelian) topological group admits an (invariant) binary closed $k$-network. In this paper we prove that the compact topological groups $S^3$ and $\SO(3)$ admit no invariant binary closed $k$-network.

math.GN↗

arXiv2013

Hereditarily supercompact spaces

A topological space $X$ is called hereditarily supercompact if each closed subspace of X is supercompact. By a combined result of Bula, Nikiel, Tuncali, Tymchatyn, and Rudin, each monotonically normal compact Hausdorff space is hereditarily supercompact. A dyadic compact space is hereditarily supercompact if and only if it is metrizable. Under (MA + not CH) each separable hereditarily supercompact space is hereditarily separable and hereditarily Lindelöf. This implies that under (MA + not CH) a scattered compact space is metrizable if and only if it is separable and hereditarily supercompact. The hereditary supercompactness is not productive: the product [0,1] x αD of the closed interval and the one-point compactification αD of a discrete space D of cardinality |D|\ge non(M) is not hereditarily supercompact (but is Rosenthal compact and uniform Eberlein compact). Moreover, under the assumption cof(M)=ω_1 the space [0,1] x αD contains a closed subspace X which is first countable and hereditarily paracompact but not supercompact.

math.GN↗

arXiv2011

Characterizing chainable, tree-like, and circle-like continua

We prove that a continuum $X$ is tree-like (resp. circle-like, chainable) if and only if for each open cover $\U_4=\{U_1,U_2,U_3,U_4\}$ of $X$ there is a $\U_4$-map $f:X\to Y$ onto a tree (resp. onto the circle, onto the interval). A continuum $X$ is an acyclic curve if and only if for each open cover $\U_3=\{U_1,U_2,U_3\}$ of $X$ there is a $\U_3$-map $f:X\to Y$ onto a tree (or the interval $[0,1]$).

math.GN↗