nLab
discrete and codiscrete topology

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Discrete and concrete objects

Contents

Definition

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

(DiscΓCodisc):TopCodiscΓDiscSet. (Disc \dashv \Gamma \dashv Codisc) : Top \stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}} Set \,.

For SSetS \in Set

For an axiomatization of this situation see codiscrete object.

Properties

Example

Let SS be a set and let (X,τ)(X,\tau) be a topological space. Then

  1. every continuous function (X,τ)Disc(S)(X,\tau) \longrightarrow Disc(S) is locally constant;

  2. every function (of sets) XCoDisc(S)X \longrightarrow CoDisc(S) is continuous.

References

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

  • William Lawvere, Functorial remarks on the general concept of chaos IMA preprint #87, 1984 (pdf)

and via footnote 1 (page 3) 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

following SGA4-1, 1.1.4.

Last revised on July 23, 2020 at 12:50:40. See the history of this page for a list of all contributions to it.