Initial topology

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

Definition in terms of open sets If

under finite intersections and arbitrary unions.

contains exactly one element, then all the open sets of the initial topology

can be characterized by the following characteristic property: A function

[4] Note that, despite looking quite similar, this is not a universal property.

[4] By the universal property of the product topology, we know that any family of continuous maps

determines a unique continuous map

[citation needed] A family of maps

separates points if and only if the associated evaluation map

and this family of maps separates points in

is a Hausdorff space if and only if these maps separate points on

is equal to the initial topology induced by the inclusion map

is equal to the initial topology induced by the canonical projections

is equal to the inverse image of the product topology on

then the evaluation map is a homeomorphism onto the subspace

is the initial topology induced by some family of maps on

separates points from closed sets in

separates points from closed sets, the space

The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.

that separates points from closed sets in

In this case, the evaluation map will be an embedding.

This uniform always exists and it is equal to the filter on

[6] It is equal to the least upper bound uniform structure of the

is also equal to the coarsest uniform structure such that the identity mappings

is the initial uniform structure induced by the mappings

[6] 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.

In the language of category theory, the initial topology construction can be described as follows.

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