Given a category of spaces equipped with a forgetful functor to Set thought of as producing for each space its underlying set of points, a codiscrete space (codiscrete object) on a set is, if it exists, the image under the right adjoint of .
The dual concept is that of discrete space.
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.