Final topology

As a final topology with respect to an imaging family is known in the mathematical branch of topology, the finest topology on a set X that makes this family of mappings from other topological spaces by X steadily. Thus the final topology results by " forward transfer " existing on the archetype spaces topological structures on the set X. This is the application of a more general concept of category theory on topological spaces, with the important "natural areas" as quotient and sum spaces in a common framework can be made. Depending on the context, then it is also called the quotient topology or sum topology.

Definition

Given a set X, a family of topological spaces and a family of mappings fi: Yi → X. A topology S on X is called a final topology with respect to the family if she has any of the following equivalent properties:

Comments

The three formulations of the definition illuminate different aspects of the final topology:

Examples

• The quotient topology is the final topology with respect to the canonical projection on the quotient space.
• The topological sum space of a family of topological spaces is the final topology on the disjoint union of the family with respect to the canonical inclusion maps. In this case, is called the final topology of the total topology.
• The combination of the sum and quotient space formation, ie the " bonding " multiple topological spaces can be made with the final topology in one step.