Recall that is a continuous map if the preimage of every open set in is open in . 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 , , the projection map is open.
If is a topological group and is a subgroup, then the projection to the coset space , where is provided with the quotient topology (making a quotient map), is open. This follows easily from the observation that if is open in , then so is
If and are open maps, then their product is also an open map.
For morphisms between locales
A continuous map of topological spaces defines a homomorphism between the frames of open sets of and . If is open, then this frame homomorphism is also a completeHeyting algebra homomorphism; the converse holds for sober spaces (maybe as long as is ?). Accordingly, we define a map of locales to be open if it is, as a frame homomorphism , a complete Heyting algebra homomorphism, i.e. it preserves arbitrary meets and the Heyting implication.