Final topology

with respect to a family of functions from topological spaces into

The dual notion is the initial topology, which for a given family of functions from a set

Explicitly, the final topology may be described as follows: The closed subsets have an analogous characterization: The family

of functions that induces the final topology on

is a proper class of functions, and the result is still well-defined in Zermelo–Fraenkel set theory.

[4] As an example, a commonly used variant of the notion of compactly generated space is defined as the final topology with respect to a proper class of functions.

[5] The important special case where the family of maps

is the final topology on the disjoint union induced by the natural injections.

is finer than (or equal to) the original topology

(See the coherent topology article for more details on this notion and more examples.)

As a particular case, one of the notions of compactly generated space can be characterized as a certain coherent topology.

is a direct system in the category Top of topological spaces and if

If a Hausdorff locally convex topological vector space

is any surjective map valued in some topological space

has the final topology induced by the maps

By the universal property of the disjoint union topology we know that given any family of continuous maps

has the final topology induced by the maps

The definition of the final topology guarantees that for every index

will be finer than (and possibly equal to) the topology

is continuous then adding these maps to the family

would necessarily be strictly coarser than the final topology

denote the space of finite sequences, where

denotes the space of all real sequences.

denote the inclusion map defined by

is strictly finer than the subspace topology induced on

that is, it is endowed with the Euclidean topology transferred to it from

is equal to the subspace topology induced on it by

is an open (respectively, closed) subset of

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

is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to SetJ then the comma category