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2 results

  • 1 - Topological Rings and Modules and Their Completions

  • Radoslav Dimitric
  • Book:
    Slenderness
    Published online:
    30 November 2018
    Print publication:
    06 December 2018, pp 41-90

  • Summary

    Linear topologies are of utmost importanceComplete objects and metrizable objects play a crucial role in characterizing slenderness.Different kinds of inversesystems, such as surjective, flabby, etc. are introduced here. Product topologies play a central role, as do Hausdorff and non-discrete topologies. Conditions ensuring that infinite sums are contained in an object are investigated.

  • Adic Topologies for the Rational Integers

  • Kevin A. Broughan
  • Journal:
    Canadian Journal of Mathematics / Volume 55 / Issue 4 / 01 August 2003
    Published online by Cambridge University Press:
    20 November 2018, pp. 711-723
    Print publication:
    01 August 2003
    • Article
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  • A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers.