# Final topology

In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set $X$ , with respect to a family of functions into $X$ , is the finest topology on $X$ that makes those functions continuous.

The dual notion is the initial topology, which for a given family of functions from a set $X$ is the coarsest topology on $X$ that makes those functions continuous.

## Definition

Given a set $X$  and a family of topological spaces $Y_{i}$  with functions

$f_{i}\colon Y_{i}\to X,$

the final topology $\tau$  on $X$  is the finest topology such that each

$f_{i}\colon Y_{i}\to (X,\tau )$

is continuous. Explicitly, the final topology may be described as follows: a subset U of X is open if and only if $f_{i}^{-1}(U)$  is open in $Y_{i}$  for each $i\in I$ .

## Examples

• 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 $\{\tau _{i}\}$  on a fixed set X, the final topology on X with respect to the functions $\operatorname {id} _{X}\colon (X,\tau _{i})\to X$  is the infimum (or meet) of the topologies $\{\tau _{i}\}$  in the lattice of topologies on X. That is, the final topology τ is the intersection of the topologies $\{\tau _{i}\}$ .
• The étalé space of a sheaf is topologized by a final topology.

## Properties

A subset of $X$  is closed/open if and only if its preimage under fi is closed/open in $Y_{i}$  for each iI.

The final topology on X can be characterized by the following characteristic property: a function $g$  from $X$  to some space $Z$  is continuous if and only if $g\circ f_{i}$  is continuous for each iI.

By the universal property of the disjoint union topology we know that given any family of continuous maps fi : YiX, there is a unique continuous map

$f\colon \coprod _{i}Y_{i}\to X.$

If the family of maps fi covers X (i.e. each x in X lies in the image of some fi) 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 (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 fi : YiX 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 (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.