# Initial topology

In general topology and related areas of mathematics, the **initial topology** (or **weak topology** or **limit topology** or **projective 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 notion is the final topology, which for a given family of functions mapping to a set is the finest topology on that makes those functions continuous.

## Definition

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

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

is continuous.

Explicitly, the initial topology is the collection of open sets generated by all sets of the form , where is an open set in for some *i* ∈ *I*, under finite intersections and arbitrary unions. The sets are often called cylinder sets.
If *I* contains exactly 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_{i}:*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, despite looking quite similar, this 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 determines a unique continuous map

This map is known as the **evaluation map**.

A family of maps is said to *separate points* in *X* if for all in *X* there exists some *i* such that . Clearly, the family 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 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 denotes 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}} that separates 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 be the functor from a discrete category to the category of topological spaces which maps . Let be the usual forgetful functor from to . The maps can then be thought of as a cone from to . That is, is an object of —the category of cones to . More precisely, this cone defines a -structured cosink in .

The forgetful functor induces a functor . The characteristic property of the initial topology is equivalent to the statement that there exists a universal morphism from to , i.e.: a terminal object in the category .

Explicitely, this consists of an object in together with a morphism such that for any object in and morphism there exists a unique morphism such that the following diagram commutes:

The assignment placing the initial topology on extends to a functor which is right adjoint to the forgetful functor . In fact, is a right-inverse to ; since is the identity functor on .

## See also

## References

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