Metrizable Subspaces of Representation Spaces
Abstract
We show that various representation spaces for continua contain a homeomorphic copy of the harmonic sequence, so even though it is not a T0 space, it contains infinite metrizable subspaces. It remains an open question whether this representation space (topologized using continuous surjections) contains a closed subspace homeomorphic to the harmonic sequence. In fact, the representation space for continua topologized using various kinds of mappings (open, confluent, or monotone mappings, for instance) can be shown to have some closed points, but it remains open in many cases whether there are more than two closed points in most instances. Examples of closed points include the pseudo-arc and the universal pseudo-solenoid. Along the way, we find a few related results about metrizable subspaces of representation spaces for continua.
Recommended Citation
W. J. Charatonik et al., "Metrizable Subspaces of Representation Spaces," Topology and its Applications, vol. 325, article no. 108351, Elsevier, Feb 2023.
The definitive version is available at https://doi.org/10.1016/j.topol.2022.108351
Department(s)
Mathematics and Statistics
Keywords and Phrases
Chainable Continua; Continuum; Hereditarily Indecomposable Continua; Representation Space; Universal Continua
International Standard Serial Number (ISSN)
0166-8641
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2023 Elsevier, All rights reserved.
Publication Date
15 Feb 2023
