Abstract
We state a simple fact relating some continuity and homomorphy properties of an internal mapping 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” mapping from a locally compact to an arbitrary topological group is “arbitrarily close” to a continuous homomorphism between them.
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