Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1972-0293002-2
MathSciNet review: 0293002
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] P. R. Halmos, Comment on the real line, Bull. Amer. Math. Soc. 50 (1944), 877-878. MR 6, 145. MR 0011301 (6:145e)
  • [2] D. N. Hawley, Compact group topologies for R, Proc. Amer. Math. Soc. 30 (1971), 566-572. MR 0281834 (43:7548)
  • [3] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #158. MR 551496 (81k:43001)
  • [4] J. L. Kelly, General topology, Van Nostrand, Princeton, N.J., 1955. MR 16, 1136. MR 0070144 (16:1136c)
  • [5] L. S. Pontrjagin, Topological groups, 2nd ed., GITTL, Moscow, 1954; English transl., Gordon and Breach, New York, 1966. MR 17, 171; MR 34 #1439. MR 0201557 (34:1439)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 22C05

Retrieve articles in all journals with MSC: 22C05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0293002-2
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

American Mathematical Society