Final topology


In general topology and related areas of mathematics, the final topology on a set, with respect to a family of functions into, is the finest topology on that makes those functions continuous.
The dual notion is the initial topology, which for a given family of functions from a set is the coarsest topology on that makes those functions continuous.

Definition

Given a set and a family of topological spaces with functions
the final topology on is the finest topology such that each
is continuous. Explicitly, the final topology may be described as follows: a subset U of X is open if and only if is open in for each.

Examples

A subset of is closed/open if and only if its preimage under fi is closed/open in for each i ∈ I.
The final topology on X can be characterized by the following characteristic property: a function from to some space is continuous if and only if is continuous for each i ∈ I.
By the universal property of the disjoint union topology we know that given any family of continuous maps fi : Yi → X, there is a unique continuous map
If the family of maps fi covers X then the map f will be a quotient map if and only if X has the final topology determined by the maps fi.

Categorical description

In the language of category theory, the final topology construction can be described as follows. Let Y be a functor from a discrete category J to the category of topological spaces Top that selects the spaces Yi for i in J. Let Δ be the diagonal functor from Top to the functor category TopJ. The comma category is then the category of cones from Y, i.e. objects in are pairs where fi : Yi → X is a family of continuous maps to X. If U is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to SetJ then the comma category is the category of all cones from UY. The final topology construction can then be described as a functor from to. This functor is left adjoint to the corresponding forgetful functor.