discrete and indiscrete topology (changes) in nLab

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Definition

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

(Disc⊣Γ⊣Codisc):Top←Codisc→Γ←DiscSet. (Disc \dashv \Gamma \dashv Codisc) : Top \stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}} Set \,.

For S∈SetS \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, it is the finest 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, which is called the indiscrete topology on SS (rarely also antidiscrete topology or codiscrete topology or trivial topology or chaotic topology (SGA4-1, 1.1.4)), it is the coarsest topology on SS; Codisc(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

The left adjoint of the discrete space functor

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

However, if we restrict the codomain of DiscDisc to locally connected spaces, then the left adjoint functor of DiscDisc does exist and it computes the set of connected components of a given locally connected space, i.e., is the π 0\pi_0 functor.

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

References

The For terminologychaotic topologyGrothendieck topologies , the terminology “chaotic” is motivated due (see to also atchaos) in

• 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 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).

In the context of Grothendieck topologies, this appears for instance in

• The Stacks Project, Example 7.6.6

following SGA4-1, 1.1.4.