nLab symmetric topological space

Contents

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

(0,1)(0,1)-Category theory

Contents

Definition

A topological space (X,τ)(X,\tau) is called a symmetric topological space or an R 0R_0-topological space if for all elements xXx \in X and yXy \in X, if for all open sets UτU \in \tau, xUx \in U implies that yUy \in U, then for all open sets UτU \in \tau, yUy \in U implies that xUx \in U. Or equivalently, if its specialization preorder is an equivalence relation.

Properties

A T 1T_1-topological space is a symmetric topological space which is also a Kolmogorov topological space. The quotient of a symmetric topological space (X,τ)(X, \tau) by its specialization order equivalence relation is a T 1T_1-topological space, the Kolmogorov quotient of XX.

preorderpartial orderequivalence relationequality
topological spaceKolmogorov topological spacesymmetric topological spaceaccessible topological space?

Created on May 23, 2023 at 21:55:12. See the history of this page for a list of all contributions to it.