nLab
discrete and codiscrete topology

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Discrete and concrete objects

Contents

Definition

The forgetful functor Γ:TopSet\Gamma : Top \to Set from Top to Set that sends any topological space to its underlying set has a left adjoint Disc:SetTopDisc : Set \to Top and a right adjoint Codisc:SetTopCodisc : Set \to Top.

(DiscΓCodisc):TopCodiscΓDiscSet. (Disc \dashv \Gamma \dashv Codisc) : Top \stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}} Set \,.

For SSetS \in Set

  • Disc(S)Disc(S) is the topological space on SS in which every subset is an open set

    this is called the discrete topology on SS, it is the finest topology on SS; Disc(S)Disc(S) is called a discrete space;

  • Codisc(S)Codisc(S) is the topological space on SS whose only open sets are the empty set and SS itself

    this is called the codiscrete topology on SS (also indiscrete topology or trivial topology), it is the coarsest topology on SS; Codisc(S)Codisc(S) is called a codiscrete space .

For an axiomatization of this situation see codiscrete object.

Properties

Example

Let SS be a set and let (X,τ)(X,\tau) be a topological space. Then

  1. every continuous function (X,τ)Disc(S)(X,\tau) \longrightarrow Disc(S) is locally constant;

  2. every function (of sets) XCoDisc(S)X \longrightarrow CoDisc(S) is continuous.

Revised on April 5, 2017 13:29:16 by Urs Schreiber (78.47.168.108)