An open map between topological spaces is one that maps open sets to open sets. It says here that
Although their definitions seem more natural, open and closed maps are much less important than continuous maps.
The usual "explanation" is that continuous functions are morphisms in Top, but why should that be the case? What does preserving pre-images have to do with being the "right" morphisms in a category? This also happens with concepts like measurability, where the "right" functions to look at are those where pre-images of measurable sets are measurable. I see no inconsistencies with defining morphisms to be e.g. open maps between objects in Top and something similar for the category of measurable sets, so why are these categories not as important? The morphisms here seem just as "structure-preserving" as the usual ones. Is there some deeper reason why it's the pre-images we should be looking at in both these cases (and more)?
