nLab
clopen set

Context

Topology

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

see also 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

Definition

A subset of a topological space is called clopen if it is both open and closed. Equivalently, a clopen set is a complemented element of the frame of open subsets, a definition which makes as good sense for locales as for spaces.

Properties

The set of clopen sets in any space forms a Boolean algebra. If the space is a Stone space, then it can be reconstructed from its Boolean algebra of clopens. On the other hand, a space is connected just when the only clopens are the empty set and the whole space.

Revised on May 2, 2012 16:41:47 by Urs Schreiber (82.113.99.15)