Initial topology

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In general topology and related areas of mathematics, the initial topology (or weak topology or limit topology or inductive topology) on a set X, with respect to a family of functions on X, is the coarsest topology on X that 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.

Definition[edit]

Given a set X and an indexed family (Yi)iI of topological spaces with functions

f_i: X \to Y_i

the initial topology τ on X is the coarsest topology on X 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. If I contains just one element, all the open sets of (X,\tau) are cylinder sets.

Examples[edit]

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

Properties[edit]

Characteristic property[edit]

The initial topology on X can be characterized by the following characteristic property:
A function g from some space Z to X is continuous if and only if f_i \circ g is continuous for each iI.

Characteristic property of the initial topology

Note that this though looking quite similar is not a universal property. A categorical description is given below

Evaluation[edit]

By the universal property of the product topology we know that any family of continuous maps fi : XYi determines a unique continuous map

f\colon X \to \prod_i Y_i\,

This map is known as the evaluation map.

A family of maps {fi: XYi} is said to separate points in X if for all xy in X there exists some i such that fi(x) ≠ fi(y). Clearly, the family {fi} 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 {fi} and this family of maps separates points in X.

Separating points from closed sets[edit]

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 {fi: XYi} 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

f_i(x)\notin \operatorname{cl}(f_i(A))

where cl denoting the closure operator.

Theorem. A family of continuous maps {fi: XYi} separates points from closed sets if and only if the cylinder sets f_i^{-1}(U), for U open in Yi, form a base for the topology on X.

It follows that whenever {fi} separates points from closed sets, the space X has the initial topology induced by the maps {fi}. 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 T0 space, then any collection of maps {fi} which separate points from closed sets in X must also separate points. In this case, the evaluation map will be an embedding.

Categorical description[edit]

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 Yj for j in J. Let U be the usual forgetful functor from Top to Set. The maps {fj} 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 UI is the identity functor on Cone(UY).

See also[edit]

References[edit]