nLab
codiscrete space

Idea

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

The dual concept is that of discrete space.

Examples

Codiscrete topological spaces

For $\Gamma :\mathrm{Top}\to \mathrm{Set}$ 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.