nLab
open map

Definition

For maps between topological spaces

A function $f:X\to Y$ between topological spaces is called open if the image of every open set in $X$ is also open in $Y$.

Recall that $f$ is a continuous map if the preimage of every open set in $Y$ is open in $X$. For defining open maps typically one restricts attention to open continuous maps, although it also makes sense to speak of open functions that are not continuous.

Examples

• For any two topological spaces $X$, $Y$, the projection map $\pi :X\times Y\to Y$ is open.

• If $G$ is a topological group and $H$ is a subgroup, then the projection to the coset space $p:G\to G/H$, where $G/H$ is provided with the quotient topology (making $p$ a quotient map), is open. This follows easily from the observation that if $U$ is open in $G$, then so is

  $$p^{-1}(p(U))=UH=\bigcup _{h\in H}Uh$$p^{-1}(p(U)) = U H = \bigcup_{h \in H} U h

• If $p:A\to B$ and $q:C\to D$ are open maps, then their product $p\times q:A\times C\to B\times D$ is also an open map.

For morphisms between locales

A continuous map $f:X\to Y$ of topological spaces defines a homomorphism $f^{*}:\mathrm{Op}(Y)\to \mathrm{Op}(X)$ between the frames of open sets of $X$ and $Y$. If $f$ is open, then this frame homomorphism is also a complete Heyting algebra homomorphism; the converse holds for sober spaces (maybe as long as $Y$ is $T_0$?). Accordingly, we define a map $f:X\to Y$ of locales to be open if it is, as a frame homomorphism $f^{*}:\mathrm{Op}(Y)\to \mathrm{Op}(X)$, a complete Heyting algebra homomorphism, i.e. it preserves arbitrary meets and the Heyting implication.

This is equivalent to saying that $f^{*}:\mathrm{Op}(Y)\to \mathrm{Op}(X)$ has a left adjoint $f_{!}$ (by the adjoint functor theorem for posets) which satisfies the Frobenius reciprocity condition that $f_{!}(U\cap f^{*}V)=f_{!}(U)\cap V$.

For geometric morphisms of toposes

Categorifying, a geometric morphism $f:X\to Y$ of toposes is an open geometric morphism if its inverse image functor $f^{*}:Y\to X$ is a Heyting functor.

For morphisms in a topos

A class $R\subset \mathrm{Mor}(ℰ)$ of morphisms in a topos $ℰ$ is called a class of open maps if it satisfies the following axioms.

1. Every isomorphism belongs to $R$;

2. The pullback of a morphism in $R$ belongs to $R$.

3. If the pullback of a morphism $f$ along an epimorphism lands in $R$, then $f$ is also in $R$.

4. For every set $S$ the canonical morphism $({\coprod }_{s\in S}*)\to *$ from the $S$-fold coproduct of the terminal object to the terminal object is in $R$.

5. For $\{X_i\stackrel{f_i}{\to }{Y}_i{\}}_{i\in I}\subset R$ then also the coproduct ${\coprod }_i{X}_i\to {\coprod }_i{Y}_i$ is in $R$.

6. If in a diagram of the form

   $$\begin{array}{ccccc}Y& & \stackrel{p}{\to }& & X\\ & {}_g\searrow & & {\swarrow }_f\\ & & B\end{array}$$\array{ Y &&\stackrel{p}{\to}&& X \\ & {}_{\mathllap{g}}\searrow && \swarrow_{\mathrlap{f}} \\ && B }

   we have that $p$ is an epimorphism and $g$ is in $R$, then $f$ is in $R$.

The class $R$ is called a class of étale maps if in addition to the axioms 1-5 above it satisfies

1. for $f:X\to Y$ in $R$ also the diagonal $Y\to Y{\times }_{X}Y$ is in $R$.

2. If in

   $$\begin{array}{ccccc}Y& & \stackrel{p}{\to }& & X\\ & {}_g\searrow & & {\swarrow }_f\\ & & B\end{array}$$\array{ Y &&\stackrel{p}{\to}&& X \\ & {}_{\mathllap{g}}\searrow && \swarrow_{\mathrlap{f}} \\ && B }

   we have that $p$ is an epimorphism, and $p,g\in R$, then $f\in R$.

For instance (JoyalMoerdijk, section 1).

References

• Andre Joyal, Ieke Moerdijk, A completeness theorem for open maps, Annals of Pure and Applied Logic 70, Issue 1, 18 November 1994, p. 51-86, MR95j:03104, doi