Metrizability of proper G-spaces and their orbit spaces

For a locally compact group G, proper G-spaces in the sense of R. Palais are studied. One of our results states that each strongly metrizable proper G-space admits a G-invariant metric (compatible with its topology) provided G is an almost connected group. This extends several results about the exis...

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Bibliographic Details
Published in:Topology and its applications 2021-09, Vol.301, p.107491, Article 107491
Main Authors: Antonyan, Sergey A., Juárez-Anguiano, Hugo, Zhang, Lili
Format: Article
Language:English
Subjects:
Dispersive dynamical system
Invariant metric
Locally compact group
Metrizable orbit space
p-Space
Proper action
Strongly metrizable space
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Summary:For a locally compact group G, proper G-spaces in the sense of R. Palais are studied. One of our results states that each strongly metrizable proper G-space admits a G-invariant metric (compatible with its topology) provided G is an almost connected group. This extends several results about the existence of invariant metrics. We also prove that if the G-orbit space X/G of a proper G-space X is metrizable, then there exists a compact subgroup H of G such that the H-orbit space X/H is metrizable too. This is applied to show that IndX=dim⁡X in this case. Another result claims that if the orbit space X/G of a proper G-space X is a paracompact p-space, then X is also such.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2020.107491