Continuous isomorphisms from<b>R</b>onto a complete abelian group

Bridges, Douglas S.; Hendtlass, Matthew

doi:10.2178/jsl/1278682208

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“…A concluding comment: our investigation of continuous group homomorphisms arose from Arno Berger asking whether the Baire category theorem was essential for the proof of COP, whose (partial) abstraction to the context of abelian groups we have analysed constructively in this paper and [7]. We still do not know the answer to Berger's question, but it is interesting that the proof of our direct constructive analogue of the (abstracted) periodicity theorem uses the "dense open sets" version of Baire's theorem, whereas our proof of a contrapositive version, in [7], uses a constructively inequivalent "union of closed sets" version.…”

Section: Wlpomentioning

confidence: 97%

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Continuous homomorphisms of R onto a compact group

Bridges¹,

Hendtlass

2010

Math. Log. Quart.

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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.

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Section: Wlpomentioning

confidence: 97%

“…We still do not know the answer to Berger's question, but it is interesting that the proof of our direct constructive analogue of the (abstracted) periodicity theorem uses the "dense open sets" version of Baire's theorem, whereas our proof of a contrapositive version, in [7], uses a constructively inequivalent "union of closed sets" version.…”

Section: Wlpomentioning

confidence: 99%