A conjugated version of the countable dense homogeneity
type: Item
instance of: scholarly article
title: A conjugated version of the countable dense homogeneity
zbMATH Open document ID: 1166.54013
author: Tadeusz Dobrowolski
published in: Topology and its Applications
publication date: 2009-03-06T00:00:00Z
review text: A separable metric space \(X\) is said to be countable dense homogeneous if for any two countable dense sets \(A\) and \(B\) in \(X\), there is a homeomorphism \(h: X\to X\) such that \(h(A)= B\). The author modifies this property considering a bijection \(\Phi: X\to X\) whose fixed point set has empty interior and such that given any two countable dense sets \(A\) and \(B\) in \(X\), there is a homeomorphism \(h: X\to X\) such that \(\Phi(h(A))= h(B)\). Among other results obtained he proves that if \(\Phi: E\to E\) is a bijection of a Banach space \(E\) such that \(\text{int}(\text{Fix}(\Phi))=\emptyset\), then, given any two countable dense sets \(A\) and \(B\) in \(E\), there exists a real-analytic diffeomorphism \(h: E\to E\) such that \(\Phi(h(A))= h(B)\). Note that \(h^{-1}(\Phi(h(A)))= B\) is equivalent to \(\Phi(h(A))= h(B)\).
reviewed by: Ludvík Janoš
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 57N17
Mathematics Subject Classification ID: 57N20
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 52A07
zbMATH DE Number: 5526277
zbMATH Keywords: countable dense homogeneity
zbMATH Keywords: real-analytic diffeomorphism
zbMATH Keywords: vector field
zbMATH Keywords: flow
zbMATH Keywords: manifold
MaRDI profile type: Publication
full work available at URL: j.topol.2008.10.011
OpenAlex ID: W2065055069
cites work: Countable dense homogeneous spaces
cites work: Q4060144
cites work: Q4098399
cites work: A Polish AR-Space with no Nontrivial Isotopy
cites work: Open embeddings of certain Banach manifolds
cites work: RISE AND FALL OF SYNTHETIC CDO MARKET: LESSONS LEARNED
cites work: Introduction to differentiable manifolds.
cites work: On Countable Dense and Strong Local Homogeneity
cites work: Measure preserving analytic diffeomorphisms of countable dense sets in $C^n$ and $R^n$
cites work: Q5557123
DOI: 10.1016/J.TOPOL.2008.10.011
description: scientific article; zbMATH DE number 5526277