nLab
discrete and codiscrete topology

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, 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), Codisc(S)Codisc(S) is called a codiscrete space .

Properties

For an axiomatization of this situation see codiscrete object.

Revised on October 11, 2012 11:50:08 by Urs Schreiber (194.78.185.20)