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

Sharpe, Kevin J.

doi:10.1090/S0002-9939-1972-0293002-2

Two properties of $R^{N}$ with a compact group topology
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by Kevin J. Sharpe

Proc. Amer. Math. Soc. 34 (1972), 267-269

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

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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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