Almost homomorphisms of compact groups

Stability of homomorphisms in the compact-open topology

“…In the paper [10] by J.Špakula and the present author, a stability result of this kind for continuous homomorphisms between compact topological groups was proved. We also remarked that it can readily be generalized to a stability result for continuous homomorphisms between arbitrary compact topological universal algebras of finite type.…”

“…In [10] (cf. also [13]), based on an example from [7], we have constructed an infinite family of pairs of compact metrizable abelian groups A, B with the following property: There is a strictly decreasing sequence of positive reals δ n converging to 0 and a sequence of continuous functions g n : A → B such that g n is a δ n -homomorphism; nevertheless, for every (not even continuous) homomorphism ϕ : A → B we have sup a∈A ρ g n (a), ϕ(a) ≥ 1, where ρ denotes an invariant metric inducing the topology of B.…”

“…Its strength lies more in the variety of its special cases than in its somewhat cumbersome general form. The prototype of Theorem 3.1 is its following special case on stability of homomorphisms between compact groups established in [10]. Let us state beforehand that O G denotes the filter of neighborhoods of the unit element in the topological group G, and a (G, H)-continuity scale is a mapping Γ : It turns out that the specific group-theoretic structure plays no role in this case.…”

10.4115/jla.v2i0.46

“…for each n and every homomorphism ϕ : G → H , and not just a continuous one (seě Spakula-Zlatoš [21] for details).…”

“…For a compact group G the two notions of stability coincide. However, as shown iň Spakula, Zlatoš [21], even for compact abelian groups G and H continuous homomorphisms G → H need not be stable. In that paper a stability result for homomorphisms between compact groups was obtained introducing a kind of controlled continuity by means of a "continuity scale."…”