- Final topology
In
general topology and related areas ofmathematics , the final topology (inductive topology or strong topology) on a set X, with respect to a family of functions into X, is thefinest topology on "X" which makes those functions continuous.Definition
Given a set X and a family of
topological space s Y_i with functions:f_i: Y_i o Xthe final topology au on X is thefinest 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 thequotient map .
* The disjoint union is the final topology with respect to the family ofcanonical injection s.
* More generally, a topological space is coherent with a family of subspaces if it has the final topology coinduced by the inclusion maps.
* Thedirect limit of anydirect 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 theinfimum (or meet) of the topologies {τ"i"} in thelattice 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 XIf 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 aquotient 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 afunctor from adiscrete category "J" to thecategory of topological spaces Top which selects the spaces "Y""i" for "i" in "J". Let Δ be thediagonal functor from Top to thefunctor category Top"J" (this functor sends each space "X" to the constant functor to "X"). Thecomma category ("Y" ↓ Δ) is then thecategory 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 theforgetful 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)"
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