This page is about the general concept. For open continuous functions see at open map.
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
A function $f : X \to Y$ between topological spaces is called an open map 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.
For any two topological spaces $X$, $Y$, the projection map $\pi \colon X \times Y \to Y$ out of the product topological space is open.
If $G$ is a topological group and $H$ is a subgroup, then the projection to the coset space $p \colon G \to G/H$, 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
If $p \colon A \to B$ and $q \colon C \to D$ are open maps, then their product $p \times q \colon A \times C \to B \times D$ is also an open map.
A continuous map $f\colon X \to Y$ of topological spaces defines a homomorphism $f^*\colon Op(Y) \to Op(X)$ 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 if $Y$ is a T-D space. Accordingly, we define a map $f\colon X \to Y$ of locales to be open if it is, as a frame homomorphism $f^*\colon Op(Y) \to Op(X)$, a complete Heyting algebra homomorphism, i.e. it preserves arbitrary meets and the Heyting implication.
This is equivalent to saying that $f^*\colon Op(Y) \to Op(X)$ has a left adjoint $f_!$ (by the adjoint functor theorem for posets) which satisfies the Frobenius reciprocity condition that $f_!(U \cap f^* V) = f_!(U) \cap V$.
In the special case that $Y$ is the one-point locale, the Frobenius reciprocity condition is automatically satisfied.
Categorifying, a geometric morphism $f\colon X \to Y$ of toposes is an open geometric morphism if its inverse image functor $f^*\colon Y \to X$ is a Heyting functor.
A class $R \subset Mor(\mathcal{E})$ of morphisms in a topos $\mathcal{E}$ is called a class of open maps if it satisfies the following axioms.
Every isomorphism belongs to $R$;
The pullback of a morphism in $R$ belongs to $R$.
If the pullback of a morphism $f$ along an epimorphism lands in $R$, then $f$ is also in $R$.
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 $R$.
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 $R$.
If in a diagram of the form
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
for $f : X \to Y$ in $R$ also the diagonal $Y \to Y \times_X Y$ is in $R$.
If in
we have that $p$ is an epimorphism, and $p, g \in R$, then $f\in R$.
For instance (JoyalMoerdijk, section 1).
Peter Johnstone, Complemented sublocales and open maps , Annals of Pure and Applied Logic 137 (2006) pp.240–255.
Peter Johnstone, Sketches of an Elephant, volume II, Oxford University Press 2002. (Section C1.5, pp.520-522)
André Joyal, Ieke Moerdijk, A completeness theorem for open maps, Annals of Pure and Applied Logic 70 no.1 (1994) pp.51-86. MR95j:03104, doi
Application to transition systems and bisimulations
Last revised on August 26, 2022 at 20:40:51. See the history of this page for a list of all contributions to it.