Group action and shift-compactness

Shift-compactness has recently been found to be the foundation stone of classical, as well as topological, regular variation; most recently it has come again to prominence in new proofs of the Effros Open Mapping Principle of group action, another ingredient of topological regular variation. Using t...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2012-08, Vol.392 (1), p.23-39
Main Authors: Miller, Harry I., Ostaszewski, A.J.
Format: Article
Language:English
Subjects:
Almost completeness
Analytic sets
Density topology
Effros Open Mapping Theorem
Fox–Mosco convergence
Group-norm
Hit-and-miss topologies
Ideal topologies
Non-commutative groups
Shift-compactness
Topological groups
Van Mill separation property
α-Favourability
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Summary:Shift-compactness has recently been found to be the foundation stone of classical, as well as topological, regular variation; most recently it has come again to prominence in new proofs of the Effros Open Mapping Principle of group action, another ingredient of topological regular variation. Using the real line under the Euclidean and density topologies as a paradigm, we develop group-action versions of shift-compactness theorems for Baire groups acting on Baire spaces under metrizable topologies and under certain refinements of these. One aim is to pursue constructive approaches rather than rely on plain Baire-category methods (so keeping more to the Banach–Mazur strategic approach). Along the way we uncover three new coarse topologies for groups of homeomorphisms. A second purpose is to establish limitations of the shift-compactness methodology.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2012.02.021