Final topology

In general topology and related areas of mathematics, the final topology (or strong topology or colimit topology or projective topology) on a set ${\displaystyle X}$, with respect to a family of functions into ${\displaystyle X}$, is the finest topology on X which makes those functions continuous.

The dual notion is the initial topology.

Definition

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

${\displaystyle f_{i}:Y_{i}\to X}$

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

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

is continuous.

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

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 {τi} on a fixed set X the final topology on X with respect to the functions idX : (X, τi) → X is the infimum (or meet) of the topologies {τi} in the lattice of topologies on X. That is, the final topology τ is the intersection of the topologies {τi}.
• The etale space of a sheaf is topologized by a final topology.

Properties

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

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

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

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 which 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 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 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.