In general topology and related areas of mathematics, the final topology (or strong, colimit, coinduced, 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 fi is closed/open in   for each iI.

The final topology on X can be characterized by the following universal property: a function   from   to some space   is continuous if and only if   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

 

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 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 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.

See alsoEdit

ReferencesEdit

  • 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)