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