nLab discrete and codiscrete topology

Context

Topology

topology

algebraic topology

Contents

Definition

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

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

For $S \in Set$

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

this is called the discrete topology on $S$, $Disc(S)$ is called a discrete space;

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

this is called the codiscrete topology on $S$ (also indiscrete topology or trivial topology), $Codisc(S)$ 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)