Final topology

In general topology and related areas of mathematics, the final topology (inductive topology or strong 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.

Definition

Given a set $X$ and a family of topological spaces $Y\_i$ with functions:$f\_i:\; Y\_i\; o\; X$the final topology $au$ on $X$ is the finest topology such that each:$f\_i:\; Y\_i\; o\; (X,\; au)$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" ∈ "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 {τ_"i"} on a fixed set "X" the final topology on "X" with respect to the functions id_"X" : ("X", τ_"i") → "X" is the infimum (or meet) of the topologies {τ_"i"} in the lattice of topologies on "X". That is, the final topology τ is the intersection of the topologies {τ_"i"}.

Properties

A subset of $X$ is closed if and only if its preimage under "f"_"i" is closed 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 $g\; circ\; f\_i$ is continuous for each "i" ∈ "I".

By the universal property of the disjoint union topology we know that given any family of continuous maps "f"_"i" : "Y"_"i" → "X" there is a unique continuous
$fcolon\; coprod\_i\; Y\_i\; o\; X$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 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 "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 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 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 also

* Initial topology

References

* Stephen Willard, "General Topology", (1970) Addison-Wesley Publishing Company, Reading Massachusetts. "(Provides a short, general introduction)"