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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

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


Author: Kevin J. Sharpe
Journal: Proc. Amer. Math. Soc. 34 (1972), 267-269
MSC: Primary 22C05
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0293002-2
PII: S 0002-9939(1972)0293002-2
Keywords: Compact group topologies for $ {R^N}$, continuous functions on $ {R^N}$ with a compact group topology, Haar measure, measurable sets of a compact group topology for $ {R^N}$
Article copyright: © Copyright 1972 American Mathematical Society