# Initial topology

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

Given a set *X* and an indexed family (*Y*_{i})_{i∈I} of topological spaces with functions

the initial topology τ on 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 is an open set in . The sets are often called cylinder sets.
If *I* contains just one element, all the open sets of are 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}}. - 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.

## Properties

### Characteristic property

The initial topology on *X* can be characterized by the following characteristic property:

A function from some space to is continuous if and only if is continuous for each *i*∈*I*.

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

### 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*.

### 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 T_{0} 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.

## 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*).

## See also

## References

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