Initial topology

An initial topology with respect to a mapping family is called in topology is the coarsest topology on a set X that makes this family of mappings from X steadily in other topological spaces. Thus the initial topology is created by " reverse transfer " existing on the image 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 spaces" such as product and subspaces in a common framework can be made.

Definition

Given a set X, a family of topological spaces ( Yi, Ti) and a family of mappings fi: X → Yi of X in the spaces Yi. A topology S on X is called the initial topology with respect to the family ( Yi, Ti, fi), if it has one of the three following equivalent properties:

The diagram on the right illustrates the universal property of the initial topology.

Comments

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

Examples

Some commonly used constructions of topological spaces can be regarded as initial topologies:

• The sub-space topology is the initial topology of the subset relative to the natural inclusion mapping.
• The product topology is the initial topology with respect to the natural projections onto the factor spaces.
• The weak topology on a normed vector space E is the initial topology with respect to the continuous linear forms on E ( ie, the topological dual space E ' of E).
• Is given on a set X is a family of topologies Ti, then the initial topology with respect to the identity (the coarsest topology which makes the identity mapping of X into all topologies, continuous) just the least upper bound of { Ti } family of the Association of topologies on X.

Categorical Description

Within the category of topological spaces can not express readily the above universal property, because you can talk to her just about continuous functions between topological spaces and not on the quantities and arbitrary functions between them ( unless you identify a lot with the discrete topology on her, etc.). However, can be characterized when a topological space carries the initial topology with respect to a family of continuous maps from this space. So be an object in a top and a family of morphisms. if and only carries the initial topology with respect to which, if any Bimorphismus with morphisms that satisfy, is an isomorphism: For such a Bimorphismus just corresponds to a bijective continuous map, ie a (not necessarily real ) coarsening, and if already exists the coarsest topology that is compatible with the pictures, so should such coarsening is an isomorphism (ie, a homeomorphism ) to be. For a single-element family whose element is a monomorphism, this condition corresponds precisely the condition for extremal monomorphism, it follows immediately that it is the topological embeddings in the extremal monomorphisms.

If you want to contrast the initial topology for a family of not necessarily continuous functions define, you have to detour to the category of sets and this set with top by the Vergissfunktors in relationship.