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Proceedings of the American Mathematical Society

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

 
 

 

Two properties of $R^{N}$ with a compact group topology


Author: Kevin J. Sharpe
Journal: Proc. Amer. Math. Soc. 34 (1972), 267-269
MSC: Primary 22C05
DOI: https://doi.org/10.1090/S0002-9939-1972-0293002-2
MathSciNet review: 0293002
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Abstract: We let $R_c^N$ be a compact additive group, and we prove that if A is an $R_c^N$-measurable set, then one of the interiors of A and $A’$ in the usual topology for ${R^N}$ (written $R_u^N$) must be void. Also we show that the only functions from ${R^N}$ to a Hausdorff space that are both $R_u^N$-continuous and $R_c^N$-measurable are the constant functions.


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Keywords: Compact group topologies for <IMG WIDTH="36" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="${R^N}$">, continuous functions on <IMG WIDTH="36" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${R^N}$"> with a compact group topology, Haar measure, measurable sets of a compact group topology for <IMG WIDTH="36" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" SRC="images/img12.gif" ALT="${R^N}$">
Article copyright: © Copyright 1972 American Mathematical Society