Contents

# 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 $S$, it is the finest 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, which is called the indiscrete topology on $S$ (rarely also antidiscrete topology or codiscrete topology or trivial topology or chaotic topology (SGA4-1, 1.1.4)), it is the coarsest topology on $S$; $Codisc(S)$ is called a indiscrete space (rarely also antidiscrete space, even more rarely codiscrete space).

For an axiomatization of this situation see codiscrete object.

## Properties

###### Example

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

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

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

### The left adjoint of the discrete space functor

The functor $Disc$ does not preserve infinite products because the infinite product topological space of discrete spaces may be nondiscrete. Thus, $Disc$ does not have a left adjoint functor.

However, if we restrict the codomain of $Disc$ to locally connected spaces, then the left adjoint functor of $Disc$ does exist and it computes the set of connected components of a given locally connected space, i.e., is the $\pi_0$ functor.

This is discussed at locally connected spaces – cohesion over sets and cosheaf of connected components.