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,

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

## References

The terminology chaotic topology is motivated (see also at chaos) in

• William Lawvere, Functorial remarks on the general concept of chaos IMA preprint #87, 1984 (pdf)

via footnote 3 in

• William Lawvere, Categories of spaces may not be generalized spaces, as exemplified by directed graphs, preprint, State University of New York at Buffalo, (1986) Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1–7 (tac:tr9, pdf)

and appears for instance in

Last revised on May 27, 2019 at 14:06:02. See the history of this page for a list of all contributions to it.