Compactly generated space

Kelley rooms or k- spaces or compact generated spaces are investigated in the mathematical discipline of topology. This is a class of spaces whose topology is to their compact subsets in close relationship and play an important role in algebraic topology for this reason.

Definition

A topological space is said Kelley space if the following conditions are met:

  • Is a Hausdorff space
  • A subset is exactly finished when the averages is completed for all compact subsets.

The terms k-space or space compactly generated are more often found in the literature, the below mentioned textbook by J. Cigler and HC Reichel uses the term Kelley room.

Examples

  • Local Compact spaces are Kelley rooms.
  • Hausdorff spaces satisfying the first countability axiom, are Kelley rooms.

Kelleyfizierung

Is a Hausdorff space, and we define a system of subsets through is completed for all compact subsets, then a finer topology on ( ie ), which makes it a Kelley space. The topological space is called the Kelleyfizierung of and is denoted by.

Is exactly then a Kelley space if applies. It can be shown that the finest topology, which generates at all compact subsets output topology.

Is a continuous map between Hausdorff spaces, it is also steadily as picture. Thus we have a functor from the category of Hausdorff spaces in the category of Kelley spaces, with continuous maps as morphisms to each. If the embedding, it must be left - adjoint.

Properties

  • A Hausdorff space and its Kelleyfizierung have the same compact sets.
  • If a Kelley - space, then for any other topological space and every picture: is continuous is continuous for all compact subsets. ( Conversely, a Hausdorff space with this property, a Kelley space; consider it. )
  • Closed subspaces of Kelley spaces are Kelley rooms again, the Kelley property is not inherited by any sub-spaces. The Arens- Fort space is not a Kelley space, but subspace of a compact and thus a Kelley space.
  • The category of Kelley spaces is a full subcategory of the category of Hausdorff spaces.
  • Is of the Kelley - rooms and a locally compact, then the product space is a Kelley space. The product of any Kelley - spaces is not a Kelley space in general. If, however, it is a product in the category of Kelley spaces.
  • Hausdorff quotient of Kelley spaces are Kelley rooms again.
  • One of the reasons why Kelley spaces are used in algebraic topology, is the following statement: Are and Kelley rooms and denotes the space of continuous functions with the compact - open topology, the following evaluation map is continuous:

Characterization

The following Chakaterisierung of Kelley spaces goes back to DE Cohen and shows that you can look at the Kelley - spaces as a generalization of locally compact spaces:

  • A Hausdorff space if and only a Kelley space if it is a quotient of a locally compact space.

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).
  • V. Srinivas: Algebraic K- Theory. 2nd edition. Birkhauser, Boston MA 1996, inter alia, ISBN 0-817-63702-8 (Progress in Mathematics 90).
  • James Dugundji: Topology. Allyn & Bacon, Boston MA 1966 ( Allyn and Bacon Series in Advanced Mathematics ) ( See also: Brown, Dubuque IA, 1989, ISBN 0-697-06889-7 ).
  • Topological space
  • Set topology
  • Algebraic Topology
Closed set Adjoint functors Arens–Fort space Product topology
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