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closed morphism

Contents

This entry is about the general concept. For the concept in topology see at closed map.

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

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Definition

Between topological spaces

A continuous function f:XYf \colon X \longrightarrow Y between topological spaces is called a closed map if the image of every closed subset in XX is also closed in YY.

Recall that ff is a continuous map if the preimage of every closed set in YY is closed in XX. For defining closed maps typically one restricts attention to closed continuous maps, although it also makes sense to speak of closed functions that are not continuous.

Between locales

Definition

A map f:XYf : X \to Y of locales is closed iff for any uO(X)u \in O(X) and vO(Y)v \in O(Y) the reciprocity relation

f *(uf *v)=f *(u)v f_*(u \vee f^*v) = f_*(u) \vee v

holds. (“\geq” is trivially always satisfied.) The map f *:O(X)O(Y)f_* : O(X) \to O(Y) is the monotone right adjoint to f *:O(Y)O(X)f^* : O(Y) \to O(X), explicitly given by

f *(u)=sup{vO(Y)|f *(v)u}. f_*(u) = \sup\{ v \in O(Y) \,|\, f^*(v) \leq u \}.

Between toposes

Definition

A geometric morphism f:f \;\colon\; \mathcal{F} \longrightarrow \mathcal{E} of toposes is closed iff for any object AA \in \mathcal{E}, the induced locale homomorphism

Sub /f *A(1)Sub (1) \mathrm{Sub}_{\mathcal{F}/f^*A}(1) \to \mathrm{Sub}_{\mathcal{E}}(1)

between the spaces of subobjects of the corresponding terminal objects is closed in the sense of def. .

Example

Let AA be an object of a topos \mathcal{E}. Then the canonical etale geometric morphism /AA\mathcal{E}/A \to A is closed iff AA fulfills the following condition, formulated in the internal language:

UA:pΩ:A(U{xA|p})(AU)p. \mathcal{E} \models \forall U \subseteq A{:} \forall p \in \Omega{:} \quad A \subseteq (U \cup \{ x \in A \,|\, p \}) \quad\Rightarrow\quad (A \subseteq U) \vee p.

Note that this condition is satisfied for any AA whatsoever if the internal language of \mathcal{E} is boolean.

Example

Let f:ABf : A \to B be a morphism in a topos \mathcal{E}. Then the induced geometric morphism /A/B\mathcal{E}/A \to \mathcal{E}/B is closed iff, in the internal language of \mathcal{E}, the fibers of ff fulfill the condition displayed at the previous example.

Remark

A geometric morphism f:f : \mathcal{F} \to \mathcal{E} is closed iff, in the internal language of \mathcal{E}, the unique locale map f *Ω ptf_* \Omega_{\mathcal{F}} \to \mathrm{pt} into the one-point space (given by the frame Ω \Omega_{\mathcal{E}}) is closed.

References

Closed maps of locales and toposes are discussed in section C3.2 of

Last revised on June 11, 2017 at 16:50:39. See the history of this page for a list of all contributions to it.