Metrizable Subspaces of Representation Spaces
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.
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
Mathematics and Statistics
Keywords and Phrases
Chainable Continua; Continuum; Hereditarily Indecomposable Continua; Representation Space; Universal Continua
International Standard Serial Number (ISSN)
Article - Journal
© 2023 Elsevier, All rights reserved.
15 Feb 2023