Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topologies of open complemented subsets

Iosif Petrakis

Abstract

We introduce cs-topologies, or topologies of open complemented subsets, as a new approach to constructive topology that preserves the duality between open and closed subsets of classical topology. Complemented subsets were used successfully by Bishop in his constructive formulation of the Daniell approach to measure and integration. Here we use complemented subsets in topology, in order to describe simultaneously an open set, the first-component of an open complemented subset, together with its given complement as a closed set, the second component of an open complemented subset. We analyse the canonical cs-topology induced by a metric, and we introduce the notion of a modulus of openness for a cs-open subset of a metric space. Pointwise and uniform continuity of functions between metric spaces are formulated with respect to the way these functions inverse open complemented subsets together with their moduli of openness. The addition of moduli of openness in the concept of a complemented open subset, given a base for the cs-topology, makes possible to define the notions of pointwise-like and uniform-like continuity of functions between csb-spaces, that is cs-spaces with a given base.

Topologies of open complemented subsets

Abstract

We introduce cs-topologies, or topologies of open complemented subsets, as a new approach to constructive topology that preserves the duality between open and closed subsets of classical topology. Complemented subsets were used successfully by Bishop in his constructive formulation of the Daniell approach to measure and integration. Here we use complemented subsets in topology, in order to describe simultaneously an open set, the first-component of an open complemented subset, together with its given complement as a closed set, the second component of an open complemented subset. We analyse the canonical cs-topology induced by a metric, and we introduce the notion of a modulus of openness for a cs-open subset of a metric space. Pointwise and uniform continuity of functions between metric spaces are formulated with respect to the way these functions inverse open complemented subsets together with their moduli of openness. The addition of moduli of openness in the concept of a complemented open subset, given a base for the cs-topology, makes possible to define the notions of pointwise-like and uniform-like continuity of functions between csb-spaces, that is cs-spaces with a given base.
Paper Structure (12 sections, 42 theorems, 210 equations)

This paper contains 12 sections, 42 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Sets with an extensional inequality and strongly extensional functions
  3. Extensional subsets
  4. Complemented subsets
  5. Complemented points
  6. The canonical $\mathrm{cs}$-topology induced by a metric
  7. Pointwise and uniformly continuous functions between metric spaces
  8. General $\mathrm{cs}$-topologies
  9. $\mathrm{cs}$-Topologies induced by bases
  10. Pointwise-like and uniformly-like continuous functions between $\textnormal{csb}$-spaces
  11. The product of two $\textnormal{csb}$-spaces and the weak $\textnormal{csb}$-topology
  12. Concluding comments

Key Result

Proposition 2.8

Let $\mathcal{X}, \mathcal{Y}, f \colon X \to Y$ in $(\textnormal{SetExtIneq}, \textnormal{StrExtFun})$ with a left inverse $g_l \colon Y \to X$ (i) $f$ is a surjection, and $g_l$ is a strong injection. (ii) If $f$ is an injection, then $g_l$ is strongly extensional. (iii) If $g_l$ is strongly extensional and surjective, then $f$ is an injection.

Theorems & Definitions (132)

  • Definition 2.1
  • Definition 2.2: Formulas in $\mathrm{BST}$
  • Definition 2.3: Strong negation in $\mathrm{BST}$
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 122 more