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Better topologies

Becker, Howard and Kechris, Alexander S. (1996) Better topologies. In: The Descriptive Set Theory of Polish Group Actions. London Mathematical Society lecture note series. No.232. Cambridge University Press , Cambridge, pp. 53-81. ISBN 9780511735264.

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This section of the book contains several results, all of which are about changing the topology on a G–space X. Given an action a : G × X → X by a Polish group G, we would like to put a new topology t on X such that: (X,t) is “nice”, e.g., metrizable or Polish; a is continuous with respect to t; and some given a–invariant set I ⊆ X is t–open (or closed or clopen). We have two different types of theorems, one type which says that there is a new topology which makes the action continuous, the other type which says that given a continuous action, there is a new, finer topology in which the action is still continuous and which makes an invariant set open. We have already seen an instance of such a result in 2.6.6, from which it follows that if G is Polish and X a Borel G–space, then we can put on X a separable metrizable topology having the same Borel structure in which the action is continuous. In §5.1, 5.2 we consider the most important case, that of Borel–measurable actions and Borel invariant sets. In §5.3, 5.4 we consider more general actions and invariant sets.

Item Type:Book Section
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Additional Information:© H. Becker & A.S. Kechris 1996.
Series Name:London Mathematical Society lecture note series
Issue or Number:232
Record Number:CaltechAUTHORS:20180809-161147539
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Official Citation:Becker, H., & Kechris, A. (1996). BETTER TOPOLOGIES. In The Descriptive Set Theory of Polish Group Actions (London Mathematical Society Lecture Note Series, pp. 53-81). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511735264.008
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:88726
Deposited By: George Porter
Deposited On:09 Aug 2018 23:21
Last Modified:16 Nov 2021 00:29

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