A stronger form of countable dense homogeneity of the plane

• Published in: Topology and its applications Vol. 246; pp. 48 - 56
• Main Authors: Morayne, Michał, Pietroń, Maciej
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2018
• Subjects: Countable dense homogeneity; Euclidean plane; 57N05; Euclidean plane; Countable dense homogeneity
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2018.06.004

Let C={C1,C2,…}, K={K1,K2,…} be countable families of subsets of the Euclidean plane R2 whose diameters tend to zero and whose closures are continua such that cl(Ci)∩cl(Cj)=∅ and cl(Ki)∩cl(Kj)=∅, for i≠j, i,j∈N. If all sets from both families C and K are ambiently homeomorphic to each other via orientation preserving automorphisms of R2, and the sets ⋃C, ⋃K are dense in R2, then there exists an automorphism f:R2→R2 of the plane such that {f[C]:C∈C}=K.