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

Definition

Given a set X and an indexed family of topological spaces with functions the initial topology \tau on X is the coarsest topology on X such that each is continuous. Definition in terms of open sets If is a family of topologies X indexed by then the of these topologies is the coarsest topology on X that is finer than each \tau_i. This topology always exists and it is equal to the topology generated by If for every i \in I, \sigma_i denotes the topology on Y_i, then is a topology on X, and the is the least upper bound topology of the I-indexed family of topologies (for i \in I). Explicitly, the initial topology is the collection of open sets generated by all sets of the form where U is an open set in Y_i for some i \in I, under finite intersections and arbitrary unions. Sets of the form f_i^{-1}(V) are often called. If I contains exactly one element, then all the open sets of the initial topology (X, \tau) are cylinder sets.

Examples

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

Properties

Characteristic property

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 i \in I. Note that, despite looking quite similar, this is not a universal property. A categorical description is given below. A filter \mathcal{B} on X converges to a point x \in X if and only if the prefilter converges to f_i(x) for every i \in I.

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 . A family of maps is said to Separating set in X if for all x \neq y in X there exists some i such that 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.

Hausdorffness

If X has the initial topology induced by and if every Y_i is Hausdorff, then X is a Hausdorff space if and only if these maps separate points on X.

Transitivity of the initial topology

If X has the initial topology induced by the I-indexed family of mappings and if for every i \in I, the topology on Y_i is the initial topology induced by some J_i-indexed family of mappings (as j ranges over J_i), then the initial topology on X induced by is equal to the initial topology induced by the -indexed family of mappings as i ranges over I and j ranges over J_i. Several important corollaries of this fact are now given. In particular, if then the subspace topology that S inherits from X is equal to the initial topology induced by the inclusion map S \to X (defined by s \mapsto s). Consequently, if X has the initial topology induced by then the subspace topology that S inherits from X is equal to the initial topology induced on S by the restrictions of the f_i to S. The product topology on \prod_i Y_i is equal to the initial topology induced by the canonical projections as i ranges over I. Consequently, the initial topology on X induced by is equal to the inverse image of the product topology on \prod_i Y_i by the evaluation map Furthermore, if the maps separate points on X then the evaluation map is a homeomorphism onto the subspace f(X) of the product space

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 separates points from closed sets in X if for all closed sets A in X and all there exists some i such that where denotes the closure operator. It follows that whenever separates points from closed sets, the space X has the initial topology induced by the maps 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 that separates points from closed sets in X must also separate points. In this case, the evaluation map will be an embedding.

Initial uniform structure

If is a family of uniform structures on X indexed by then the of is the coarsest uniform structure on X that is finer than each This uniform always exists and it is equal to the filter on X \times X generated by the filter subbase If \tau_i is the topology on X induced by the uniform structure then the topology on X associated with least upper bound uniform structure is equal to the least upper bound topology of Now suppose that is a family of maps and for every i \in I, let be a uniform structure on Y_i. Then the is the unique coarsest uniform structure \mathcal{U} on X making all uniformly continuous. It is equal to the least upper bound uniform structure of the I-indexed family of uniform structures (for i \in I). The topology on X induced by \mathcal{U} is the coarsest topology on X such that every is continuous. The initial uniform structure \mathcal{U} is also equal to the coarsest uniform structure such that the identity mappings are uniformly continuous. Hausdorffness: The topology on X induced by the initial uniform structure \mathcal{U} is Hausdorff if and only if for whenever x, y \in X are distinct (x \neq y) then there exists some i \in I and some entourage of Y_i such that Furthermore, if for every index i \in I, the topology on Y_i induced by is Hausdorff then the topology on X induced by the initial uniform structure \mathcal{U} is Hausdorff if and only if the maps separate points on X (or equivalently, if and only if the evaluation map is injective) Uniform continuity: If \mathcal{U} is the initial uniform structure induced by the mappings then a function g from some uniform space Z into is uniformly continuous if and only if is uniformly continuous for each i \in I. Cauchy filter: A filter \mathcal{B} on X is a Cauchy filter on if and only if is a Cauchy prefilter on Y_i for every i \in I. Transitivity of the initial uniform structure: If the word "topology" is replaced with "uniform structure" in the statement of "transitivity of the initial topology" given above, then the resulting statement will also be true.

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 which maps. Let U be the usual forgetful functor from to. The maps can then be thought of as a cone from X to UY. That is, (X,f) is an object of —the category of cones to UY. More precisely, this cone (X,f) defines a U-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 \bar{U} to (X,f); that is, a terminal object in the category Explicitly, this consists of an object I(X,f) in together with a morphism such that for any object (Z,g) in and morphism there exists a unique morphism such that the following diagram commutes: The assignment placing the initial topology on X extends to a functor which is right adjoint to the forgetful functor \bar{U}. In fact, I is a right-inverse to \bar{U}; since \bar{U}I is the identity functor on