nLab unified topological space

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

A notion of topological space which comes with predefined notions of neighbourhood, apartness, and nearness, without having to define any of them in terms of the others.

Definition

A unified topological space is a set AA with a unified topology - three relations \ll, \bowtie, and \approx between AA and its power set 𝒫(A)\mathcal{P}(A), such that for all elements xx of AA and subsets UU and VV of AA,

  • \ll is a topology in the usual sense:

    • if xUx \ll U, then xUx \in U

    • if xUx \ll U and UVU \subseteq V, then xVx \ll V

    • xAx \ll A

    • if xUx \ll U and xVx \ll V, then xUVx \ll U \cap V

    • if xUx \ll U, then x{yA|yU}x \ll \{y \in A \vert y \ll U\}

  • \bowtie satisfies the following apartness axioms:

    • if xUx \bowtie U, then xUx \in U is false

    • if xUx \bowtie U and UVU \subseteq V, then xVx \bowtie V

    • xx \bowtie \emptyset

    • if xUx \bowtie U and xVx \bowtie V, then xUVx \bowtie U \cup V

    • if xUx \bowtie U, then x{yA|yU}x \bowtie \{y \in A \vert y \approx U\}

  • \approx satisfies the following closure space axioms:

    • if xUx \in U, then xUx \approx U

    • if xUx \approx U and UVU \subseteq V, then xVx \approx V

    • xx \approx \emptyset is false

    • if xUVx \approx U \cup V and xUx \approx U, then xVx \approx V

    • if x{yA|yU}x \approx \{y \in A \vert y \approx U\}, then xUx \approx U

  • The following compatibility condition holds:

    • if xUx \ll U and xVx \approx V, then there exists an element yy of AA such that yUVy \in U \cap V.

 References

Unified topologies were defined in definition 10.17 of:

Created on September 6, 2024 at 11:58:07. See the history of this page for a list of all contributions to it.