# Final topology

In general topology and related areas of mathematics, the **final topology** (or **coinduced**,^{[1]} **strong**, **colimit**, or **inductive** 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.

## DefinitionEdit

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 .

## ExamplesEdit

- The quotient topology is the final topology on the quotient space with respect to the quotient map.
- The disjoint union is the final topology with respect to the family of canonical injections.
- More generally, a topological space is coherent with a family of subspaces if it has the final topology coinduced by the inclusion maps.
- The direct limit of any direct system of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms.
- Given a family of topologies on a fixed set
*X*, the final topology on*X*with respect to the functions is the infimum (or meet) of the topologies in the lattice of topologies on*X*. That is, the final topology τ is the intersection of the topologies . - The étalé space of a sheaf is topologized by a final topology.

## PropertiesEdit

A subset of is closed/open if and only if its preimage under *f*_{i} 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 *f*_{i} : *Y*_{i} → *X*, there is a unique continuous map

If the family of maps *f*_{i} *covers* *X* (i.e. each *x* in *X* lies in the image of some *f*_{i}) then the map *f* will be a quotient map if and only if *X* has the final topology determined by the maps *f*_{i}.

## Categorical descriptionEdit

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 *Y*_{i} for *i* in *J*. Let Δ be the diagonal functor from **Top** to the functor category **Top**^{J} (this functor sends each space *X* to the constant functor to *X*). The comma category (*Y* ↓ Δ) is then the category of co-cones from *Y*, i.e. objects in (*Y* ↓ Δ) are pairs (*X*, *f*) where *f*_{i} : *Y*_{i} → *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 **Set**^{J} then the comma category (*UY* ↓ Δ′) is the category of all co-cones from *UY*. The final topology construction can then be described as a functor from (*UY* ↓ Δ′) to (*Y* ↓ Δ). This functor is left adjoint to the corresponding forgetful functor.

## See alsoEdit

## ReferencesEdit

**^**Singh, Tej Bahadur (May 5, 2013). "Elements of Topology".*Books.Google.com*. CRC Press. Retrieved July 21, 2020.

## SourcesEdit

- Willard, Stephen (1970).
*General Topology*. Addison-Wesley Series in Mathematics. Reading, MA: Addison-Wesley. Zbl 0205.26601..*(Provides a short, general introduction in section 9 and Exercise 9H)*