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Locally hulled topologies

Eugene Bilokopytov

Abstract

We present a general result about generating group topologies by pseudo-norms. Namely, we show that if a topology has a base of sets which are closed in a certain sense, then it can be generated by a collection of pseudo-norms such that the balls in these pseudo-norms are also closed in the same sense. The examples include linear and locally convex topologies on vector spaces, locally solid and Fatou topologies on vector lattices and Fréchet-Nikodým topologies on Boolean algebras.

Locally hulled topologies

Abstract

We present a general result about generating group topologies by pseudo-norms. Namely, we show that if a topology has a base of sets which are closed in a certain sense, then it can be generated by a collection of pseudo-norms such that the balls in these pseudo-norms are also closed in the same sense. The examples include linear and locally convex topologies on vector spaces, locally solid and Fatou topologies on vector lattices and Fréchet-Nikodým topologies on Boolean algebras.

Paper Structure

This paper contains 6 sections, 20 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Hull structures
  3. Additive topologies on monoids
  4. Group topologies
  5. Linear topologies
  6. Applications

Key Result

Proposition 2.1

Assume that $\mathcal{R}$ is a $1$-algebraic hull structure on $X$. A topology $\tau$ on $X$ is a $\mathcal{R}$-topology if and only if $x_{p}\xrightarrow[]{\tau}0_{X}$$\Rightarrow$$y_{p}\xrightarrow[]{\tau}0_{X}$, where $y_{p}\in\left\{x_{p}\right\}^{\mathcal{R}}$, for every $p\in P$.

Theorems & Definitions (49)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Example 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Theorem 3.1
  • proof
  • ...and 39 more