discrete and codiscrete topology




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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

  • Disc(S)Disc(S) is the topological space on SS in which every subset is an open set,

    this is called the discrete topology on SS, it is the finest topology on SS; Disc(S)Disc(S) is called a discrete space;

  • Codisc(S)Codisc(S) is the topological space on SS whose only open sets are the empty set and SS itself,

    this is called the codiscrete topology on SS (also indiscrete topology or trivial topology or chaotic topology), it is the coarsest topology on SS; Codisc(S)Codisc(S) is called a codiscrete space.

For an axiomatization of this situation see codiscrete object.



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.


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)

via footnote 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)

and appears for instance in

Last revised on May 27, 2019 at 14:06:02. See the history of this page for a list of all contributions to it.