Abstract
We state a simple fact relating some continuity and homomorphy properties of an internal map** between nonstandard extensions of topological groups and the nonstandard extension of the “observable trace” of the map, which can be interpreted as a kind of stability principle. This leads to a strengthening of two formerly proved (standard) stability results (a global one and a local one) along with simplifying their proofs. We show that every “sufficiently continuous,” “reasonably bounded” and “sufficiently homomorphic” map** from a locally compact to an arbitrary topological group is “arbitrarily close” to a continuous homomorphism between them.
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Research supported by the Grant no. 1/0608/13 of the Slovak grant agency VEGA.
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Sládek, F., Zlatoš, P. A local stability principle for continuous group homomorphisms in nonstandard setting. Aequat. Math. 89, 991–1001 (2015). https://doi.org/10.1007/s00010-014-0301-7
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DOI: https://doi.org/10.1007/s00010-014-0301-7