In topology and related areas of mathematics, an induced topology on a topological space is a topology which is "optimal" for some function from/to this topological space.
Let be sets, .
If is a topology on , then a topology coinduced on by is .
If is a topology on , then a topology induced on by is .
The easy way to remember the definitions above is to notice that finding an inverse image is used in both. This is because inverse image preserves union and intersection. Finding a direct image does not preserve intersection in general. Here is an example where this becomes a hurdle. Consider a set with a topology , a set and a function such that . A set of subsets is not a topology, because but .
There are equivalent definitions below.
A topology induced on by is the finest topology such that is continuous . This is a particular case of the final topology on .
A topology induced on by is the coarsest topology such that is continuous . This is a particular case of the initial topology on .