Initial topology

In general 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

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

$f_{i}:X\to Y_{i}$

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

$f_{i}:(X,\tau )\to Y_{i}$

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 $Y_{i}$. The sets $f_{i}^{-1}(U)$ are often called cylinder sets.

Examples

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

Composition

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 $f_{i}\circ g$ is continuous for each i ∈ I.

Characteristic property of the initial topology

Evaluation

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

$f\colon X\to \prod _{i}Y_{i}\,$

This map is known as the evaluation map.

Embedding

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 evaluation map f will be a topological embedding if and only if X has the initial topology determined by the maps f_i.

Base

A family of maps {f_i} separates points from closed sets in X if for all closed A in X and all $x\notin A$, there exists some f_i such that $f_{i}(x)\notin \operatorname {cl} (f_{i}(A))$ (with cl denoting the closure operator).

Then a family of continuous maps {f_i} separates points from closed sets if and only if the sets $f_{i}^{-1}(U)$, for U an open set in Y_i, form a base for the topology on X. As a corollary, this topology is the weak topology induced by the maps {f_i}.

T₁ space

If the space X is a T₁ space, and the {f_i} are a collection of maps that separate points from closed sets, then the evaluation map is an embedding.

Categorical description

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 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

• Final topology

References

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