nLab symmetric topological space

Redirected from "R0-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.

 Examples

Every set XX with its power set 𝒫(X)\mathcal{P}(X) is a symmetric topological space with respect to the specialization order equivalence relation defined as

xyP𝒫(X).P(x)P(y)x \equiv y \coloneqq \forall P \in \mathcal{P}(X).P(x) \iff P(y)

for all xXx \in X and yXy \in X.

preorderpartial orderequivalence relationequality
topological spaceKolmogorov topological spacesymmetric topological spaceaccessible topological space

Last revised on July 16, 2023 at 20:22:14. See the history of this page for a list of all contributions to it.