nLab
codiscrete space

Contents

Idea

Given a category SpSp of spaces equipped with a forgetful functor Γ:SpSet\Gamma : Sp \to Set to Set thought of as producing for each space its underlying set of points, a codiscrete space (codiscrete object) Codisc(S)Codisc(S) on a set SS is, if it exists, the image under the right adjoint Codisc:SetSpCodisc : Set \to Sp of Γ\Gamma.

Sometimes the codiscrete topology is also called the chaotic topology.

The dual concept is that of discrete space. For their relation see at discrete and codiscrete topology.

Examples

Codiscrete topological spaces

For Γ:TopSet\Gamma : Top \to Set the obvious forgetful functor from Top, a codiscrete space is a set with codiscrete topology.

Codiscrete cohesive spaces

A general axiomatization of the notion of space is as an object in a cohesive topos. This comes by definition with an underlying-set-functor (or similar) and a left adjoint that produces discrete cohesive structure. See there for details.

References

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

via footnote 3 in

and appears for instance in