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.

The dual concept is that of discrete space.

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.

Revised on November 23, 2011 17:25:17 by Urs Schreiber (131.174.40.49)