Continuous homomorphisms of R onto a compact group

Abstract

It is shown within Bishop's constructive mathematics that, under one extra, classically automatic, hypothesis, a continuous homomorphism from R onto a compact metric abelian group is periodic, but that the existence of the minimum value of the period is not derivable.In this paper, which is written within the framework of Bishop's constructive mathematics (BISH), 1) we consider a partial abstraction of the well-known classical result COPEvery compact orbit of a dynamical system is periodic [12].The abstraction is the following: Theorem 1 Let θ be a continuous homomorphism of R onto a compact (metric) abelian group G, such thatis open. Then θ is periodic, in the sense that there exists τ > 0 such that θ(τ) = 0.Here we use standard additive notation for abelian group operations. By a metric abelian group we mean an abelian group G equipped with a metric ρ, such that the mapping (x, y)y − x is pointwise continuous at (0, 0) ∈ G × G, and uniformly continuous on compact subsets of G × G. The mappings x −x and (x, y)x + y are then pointwise continuous throughout their domains, and uniformly continuous on compact subsets of their domains. Moreover, for each positive integer n, the mapping x nx is continuous at 0. Note that if G is locally compact -that is, every bounded subset of G is contained in a compact subset of G -then the pointwise continuity of the mapping (x, y)y − x is a consequence of its uniform continuity on compact sets. We say that a homomorphism θ of the abelian group R into a metric abelian group G is continuous if it is uniformly continuous on each compact (or, equivalently, on each bounded) subset of R.Although we know of no reference for Theorem 1 in the literature, we believe that the following argument, based on the standard classical one used to prove COP, would be the natural one for the classical mathematician to use. Under the hypotheses of Theorem 1, first observe that for each positive integer n, since θ is continuous, the closed subsetof G is compact. Since C 1 ⊃ C 2 ⊃ · · · , the set n 1 C n is nonempty; from which it is not hard to deduce that each C n is dense in G. On the other hand, by the continuity of θ, the sets θ[−n, n] are compact and hence closed. Since G is complete and equals the union of the sets θ[−n, n] (n 1), it follows from the Baire category theorem that the interior of θ[−N, N] is nonempty for some N. We can therefore find t 1 , t 2 such that |t 1 | N, t 2 N + 1, and θ(t 1 ) = θ(t 2 ). Setting τ = t 2 − t 1 , with very little more work we see that θ(τ) = 0.

Continuous isomorphisms fromRonto a complete abelian group

“…Accordingly, given / € R such that 9{t) = 0, suppose that t ^ 0; in order to derive a contradiction, we may assume that t > 0. Then, by Proposition 10 of [7], the set #[0, t], which is totally bounded since 9 is uniformly continuous on [0, t], is dense in G. Being complete, G is therefore compact-a contradiction. We conclude that -<(t ^ 0), from which it follows that t = 0.…”

“…In the context of abelian groups, one abstraction of COP is that if 6 is a continuous homomorphism of R onto a compact (metric) abelian group G, then there exists T > 0 such that 0(z) = 0. That abstraction is considered in [7]; the main result of the present paper is a contrapositive of it: namely, THEOREM 1. Let 9 be a continuous, one-one homomorphism of R onto a complete {metric) abelian group (G,p), and let Suppose that Si is weakly located at 0.…”

“…Since we explain in the introduction to [7] where the natural classical proof of the contrapositive of Theorem 1 breaks down constructively, we now proceed, without further comment, to the development of the results of the present paper, which has three more sections: one dealing with a number of results about one-one and injective mappings that are of interest in their own right, the second covering some aspects of locatedness and a version of the Baire category theorem, and the third applying those results to the proof of Theorem 1 and its preliminary lemmas.…”