Induced topology


In topology and related areas of mathematics, an induced topology on a topological space is a topology that makes a given function or collection of functions continuous from this topological space.
A coinduced topology or final topology makes the given collection of functions continuous to this topological space.

Definition

The case of just one function

Let be sets,.
If is a topology on, then the topology coinduced on by is.
If is a topology on, then the 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.
The topology coinduced on by is the finest topology such that is continuous. This is a particular case of the final topology on.
The topology induced on by is the coarsest topology such that is continuous. This is a particular case of the initial topology on.

General case

Given a set X and an indexed family i∈I of topological spaces with functions
the topology on induced by these functions is the coarsest topology on X such that each
is continuous.
Explicitly, the induced topology is the collection of open sets generated by all sets of the form, where is an open set in for some i ∈ I, under finite intersections and arbitrary unions. The sets are often called cylinder sets.
If I contains exactly one element, all the open sets of are cylinder sets.

Examples