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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Relating group topologies by their continuous points


Author: Kevin J. Sharpe
Journal: Proc. Amer. Math. Soc. 57 (1976), 179-182
MSC: Primary 22B99
DOI: https://doi.org/10.1090/S0002-9939-1976-0422488-9
MathSciNet review: 0422488
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Abstract: Let $ x$ be a point in a topological group $ G$, and for each integer $ n$, let $ (1/n)x$ be the set $ \{ y:ny = x\} $ in $ G$. Then I call $ x$ a continuous point if for positive integers $ n$, the subsets $ (1/n)x$ are nonvoid and eventually intersect each neighbourhood of the identity 0. I prove the following result and from it three corollaries. Let $ G$ be a divisible abelian group such that $ (1/n)0 = \{ 0\} $ for some integer $ n \geqslant 2$. Suppose there are two group topologies $ {\mathcal{A}_1}$ and $ {\mathcal{A}_2}$ defined on $ G$ and that $ G$ is $ {\mathcal{A}_2}$-locally compact and $ \sigma $-compact, and define $ {\omega _2}$ to be the outer measure derived from the Haar measure $ {\mu _2}$ on $ (G,{\mathcal{A}_2})$. Also suppose that the ratio of the $ {\mathcal{A}_2}$-measure of $ \{ nx:x \in A\} $ to the $ {\mathcal{A}_2}$-measure of $ A$, for any $ {\mathcal{A}_2}$-Borel-measurable set $ A$ (the ratio is the same for any such $ A$ with finite measure), does not exceed 1. Then for each $ {\mathcal{A}_2}$-Borel-measurable set $ A$ with nonvoid $ {\mathcal{A}_1}$-interior, $ {\mu _2}(A) \geqslant {\omega _2}({W_1}),{W_1}$ being the subgroup of all points in $ G$ which are $ {\mathcal{A}_1}$-continuous.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0422488-9
Keywords: Continuous points, topological group, Haar measure, Hawley property, uniquely rooted group
Article copyright: © Copyright 1976 American Mathematical Society