CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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.
For any two topological spaces $X$, $Y$, the projection map $\pi \colon X \times Y \to Y$ is open.
If $G$ is a topological group and $H$ is a subgroup, then the projection to the coset space $p \colon G \to G/H$, 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 for sober spaces (maybe as long as $Y$ is $T_0$?). 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 $f_!(U \cap f^* V) = f_!(U) \cap V$.
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).
An application: