Let G be a group which acts by homeomorphisms on a metric space X. We say the action of G is locally moving on X if for every open U ⊆ X there is a g ∈ G such that g↾X ≠ Id while g↾(X∖U) = Id. We prove the following theorem: Theorem A. Let X,Y be completely metrizable spaces and let G be a group which acts on X and Y with locally moving actions. If the orbits of the action of G on X are of the second category in X and the orbits of the action of G on Y are of the second category in Y, then X and Y are homeomorphic. A particular case of Theorem A gives a positive answer to a question of M. Rubin and J. van Mill who asked whether X and Y are homeomorphic whenever G is strongly locally homogeneous on X and Y.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-1-1,
author = {Edmund Ben-Ami},
title = {A reconstruction theorem for locally moving groups acting on completely metrizable spaces},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {1-8},
zbl = {1202.54024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-1-1}
}
Edmund Ben-Ami. A reconstruction theorem for locally moving groups acting on completely metrizable spaces. Fundamenta Mathematicae, Tome 209 (2010) pp. 1-8. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-1-1/