Supriya Ghosh (Editor)

Final topology

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

Contents

The dual notion is the initial topology.

Definition

Given a set X and a family of topological spaces Y i with functions

f i : Y i → X

the final topology τ on X is the finest topology such that each

f i : Y i → ( X , τ )

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 Yi for each i ∈ 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 {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 X is closed/open if and only if its preimage under fi is closed/open in Y i for each i ∈ I.

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

    f : ∐ i Y i → 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 : 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 (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.

    References

    Final topology Wikipedia
    (Text) CC BY-SA