Compact-open topology

The compact - open topology is a topology - short KO considered in the mathematical branch of topology structure on function spaces of continuous functions. And topological spaces are namely, the continuous maps are the structure preserving mappings. Therefore, it is natural to re- equip the set of all continuous functions with a topology. Among the many ways to do that, the compact open topology has been found to be particularly suitable.

The mathematician RH Fox (1945 ) and Richard Friederich Arens (1946 ) defined as the first, this topology and studied systematically.

Definition

Let and be topological spaces. Is compact and open, so is.

The compact -open topology is that of all sets of the form, compact, open, generated topology, ie, the open sets of this topology are arbitrary unions of finite intersections of such sets.

The quantities compact, open, thus forming a sub-base of the compact -open topology. This topology is often abbreviated (English compact- open), then referred to the room, which is equipped with the compact -open topology.

Properties

Below are and topological spaces.

Separation axioms

If Y is T0- space, T1 - space, Hausdorff space, regular room or a completely regular space, the same separation axiom is sufficient.

The evaluation map

For each non- empty subset has the evaluation map. Is any topology, so that constantly is ( thereby carries out the product and topology given to the topology ) as is, ie, the relative compact open topology on coarser than. In an important special case of the evaluation map is continuous if you know the relative compact -open topology; we have:

Locally compact and any topological space, the compact open topology on each subset the coarsest topology which makes the evaluation image continuous.

Composition

Let and be locally compact, is a third topological space. Then the composition mapping

Steadily.

Compact convergence

Be locally compact, uniform space. Then correct the compact -open topology consistent with the topology of compact convergence.

Application

As a typical application in algebraic topology here the recursive definition of higher homotopy groups is presented. It is a topological space with a marked point. With the fundamental group is denoted the base point. For the definition of higher homotopy groups, consider by the space of all continuous maps of the unit square that represent the boundary of the unit square to the base point. Denoting the constant function that maps the unit square to the point, and you know with with the relative compact -open topology, so the pair is a topological space with an excellent point.

We now define and general recursively.

Swell

• Johann Cigler, Hans- Christian Reichel: Topology. A basic lecture. Bibliographical Institute, Mannheim, inter alia, 1978, ISBN 3-411-00121-6 ( BI university paperbacks 121).
• Horst Schubert: topology. An Introduction. Teubner, Stuttgart, 1964 ( Teubner mathematical guidance. ZDB - ID 259127-3 ), ( 4th ed. Teubner, Stuttgart 1975, ISBN 3-519-12200-6 ).