Initial topology

From Wikipedia, the free encyclopedia

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.

The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.

The dual construction is called the final topology.

Contents 1 Definition 2 Examples 3 Properties 3.1 Characteristic property 3.2 Evaluation 3.3 Separating points from closed sets 4 Categorical description 5 See also 6 References

[edit] Definition

Given a set X and an indexed family (Y_i)_i∈I of topological spaces with functions

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

is continuous.

Explicitly, the initial topology may be described as the topology generated by sets of the form , where U is an open set in Y_i. The sets are often called cylinder sets.

[edit] 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}.
• A topological space is completely regular if and only if it has the initial topology with respect to its family of (bounded) real-valued continuous functions.
• Every topological space X has the initial topology with respect to the family of continuous functions from X to the Sierpiński space.

[edit] Properties

[edit] Characteristic property

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 is continuous for each i ∈ I.

[edit] 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

This map is known as the evaluation map.

A family of maps {f_i: X → Y_i} is said to separate points in X if for all x ≠ y in X there exists some i such that f_i(x) ≠ f_i(y). Clearly, the family {f_i} separates points if and only if the associated evaluation map f is injective.

The evaluation map f will be a topological embedding if and only if X has the initial topology determined by the maps {f_i} and this family of maps separates points in X.

[edit] Separating points from closed sets

If a space X comes equipped with a topology, it is often useful to know whether or not the topology on X is the initial topology induced by some family of maps on X. This section gives a sufficient (but not necessary) condition.

A family of maps {f_i: X → Y_i} separates points from closed sets in X if for all closed sets A in X and all x not in A, there exists some i such that

where cl denoting the closure operator.

Theorem. A family of continuous maps {f_i: X → Y_i} separates points from closed sets if and only if the cylinder sets , for U open in Y_i, form a base for the topology on X.

It follows that whenever {f_i} separates points from closed sets, the space X has the initial topology induced by the maps {f_i}. The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.

If the space X is a T₁ space, then any collection of maps {f_i} which separate points from closed sets in X must also separate points. In this case, the evaluation map will be an embedding.

[edit] Categorical description

In the language of category theory, the initial topology construction can be described as follows. Let Y be the functor from a discrete category J to the category of topological spaces Top which selects the spaces Y_j for j in J. Let U be the usual forgetful functor from Top to Set. The maps {f_j} can then be thought of as a cone from X to UY. That is, (X, f) is an object of Cone(UY)—the category of cones to UY.

The characteristic property of the initial topology is equivalent to the statement that there exists a universal morphism from the forgetful functor

U′ : Cone(Y) → Cone(UY)

to the cone (X, f). By placing the initial topology on X we therefore obtain a functor

I : Cone(UY) → Cone(Y)

which is right adjoint to the forgetful functor U′. In fact, I is a right-inverse to U′ since U′I is the identity functor on Cone(UY).

[edit] See also

• Final topology

[edit] References

• Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6 (Dover edition). 
• Initial topology on PlanetMath
• Product topology and subspace topology on PlanetMath