nLab discrete and indiscrete topology

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.

References

For Grothendieck topologies, the terminology “chaotic” is due to

• SGA4-1, 1.1.4 (1972)

reviewed, e.g., in:

• The Stacks Project, Example 7.6.6

Conceptualization of the terminology via right adjoints to forgetful functors (see also at chaos) is due to

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

and via footnote 1 (page 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).