Locally quasi-convex topologies on the group of the integers

الأصل · EN

The most natural group topology on is the discrete one. There are other well-known group topologies on, like the p-adic, defined for any prime number p. It is also an important group topology the weak topology with respect to the group of homomorphisms from to the unit circle of the complex plane; that is, the one defined by the characters and which is known as "the Bohr topology" on. In tesislorenzo, it is proved that taking as a neighbourhood basis at 0 the subsets {Wₙ n∈}, defined by Wₙ:={k∈∀ x∈ S, k· x∈[-1/4n,1/4n]+}, where S is a quasi-convex sequence in, a group topology on is obtained, τₛ. We know that the topology τₛ is metrizable and locally quasi-convex. In this monograph we characterize convergent sequences in τₛ, for some S⊂. We give sufficient conditions on the elements of in order that they belong to a neighbourhood Wₙ. For a fixed sequence (bₙ) of natural numbers, restricted to mild conditions, we have considered the "linear topology associated" whose neighbourhood basis at 0 is {bₙ n∈} and the "topology of uniform convergence" on S={1/bₙ+ n∈}⊂. We make a comparative study between both classes of topologies.