In topology and related areas of mathematics, the initial topology (projective topology or weak topology) on a set $X$, with respect to a family of functions on $X$, is the coarsest topology on X which makes those functions continuous.

Definition

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Given a set X and a family of topological spaces Y_i with functions

$f\_i:\; X\; \backslash to\; Y\_i$

the initial topology τ on $X$ is the coarsest topology such that each

$f\_i:\; (X,\backslash tau)\; \backslash to\; Y\_i$

is continuous.

Explicitly, the initial topology may be described as the topology generated by sets of the form $f\_i^\{-1\}(U)$, where $U$ is an open set in $Y\_i$.

Examples

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Several topological constructions can be regarded as special cases of the initial topology.

• The subspace topology is the initial topology on the subspace with respect to the inclusion map.
• The product topology is the initial topology with respect to the family of projection maps.
• The inverse limit of any inverse system of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms.
• The weak topology on a locally convex space is the initial topology with respect to the continuous linear forms of its dual space.
• Given a family of topologies {τ_i} on a fixed set X the initial topology on X with respect to the functions id_X : X → (X, τ_i) is the supremum (or join) of the topologies {τ_i} in the lattice of topologies on X. That is, the initial topology τ is the topology generated by the union of the topologies {τ_i}.

Properties

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The initial topology on X can be characterized by the following universal property: a function $g$ from some space $Z$ to $X$ is continuous if and only if $f\_i\; \backslash circ\; g$ is continuous for each i ∈ I.

By the universal property of the product topology we know that any family of continuous maps f_i : X → Y_i determines a unique continuous map

$f\backslash colon\; X\; \backslash to\; \backslash prod\_i\; Y\_i\backslash ,$

If the family of maps {f_i} separates points in X (i.e. for all x ≠ y in X there exists some f_i such that f_i(x) ≠ f_i(y)) then the map f will be a topological embedding if and only if X has the initial topology determined by the maps f_i.

In the language of category theory, the initial 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 (Δ ↓ Y) is then the category of all cones to Y, i.e. objects in (Δ ↓ Y) are pairs (X, f) where f_i : X → Y_i is a family of continuous maps on 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 (Δ′ ↓ UY) is the category of all cones to UY. The initial topology construction can then be described as a functor from (Δ′ ↓ UY) to (Δ ↓ Y). This functor is right adjoint to the corresponding forgetful functor.

See also

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• Final topology

Topology