Kelley room
Kelley spaces or k-spaces or compactly generated spaces are examined in the mathematical discipline of topology . It is a class of spaces whose topology is closely related to their compact subsets and which therefore play an important role in algebraic topology .
definition
A topological space is called a Kelley space if the following conditions are met:
- is a Hausdorff room
- A subset is closed if and only if the averages for all compact subsets is closed.
The terms k-space or compactly generated space are found more frequently in the literature, the textbook by J. Cigler and HC Reichel mentioned below uses the term Kelley space.
Examples
- Local compact rooms are Kelley rooms.
- Hausdorff spaces that satisfy the first countability axiom are Kelley spaces.
Kelleyfication
If a Hausdorff space is defined and a system of subsets is defined by is closed for all compact subsets , then there is a finer topology on (that is, ) that makes a Kelley space. The topological space is called the Kelleyfication of and is denoted by.
is a Kelley space if and only if holds. It can be shown that the finest topology creates the initial topology on all compact subsets.
If there is a continuous mapping between Hausdorff spaces, then it is also continuous as a mapping . We thus have a functor from the category of Hausdorff spaces to the category of Kelley spaces, each with the continuous mappings as morphisms. If embedding is left-adjoint to .
properties
- A Hausdorff space and its Kelleyfication have the same compact sets.
- If a Kelley space, then for every other topological space and every mapping : is continuous is continuous for all compact subsets . (Conversely, a Hausdorff room with this property is a Kelley room; see for this .)
- Closed subspaces of Kelley spaces are again Kelley spaces; the Kelley property is not inherited by any subspaces. The Arens Fort room is not a Kelley room, but a sub-room of a compact and thus a Kelley room.
- The category of Kelley rooms is a full sub- category of the Hausdorff rooms category.
- If one of the Kelley spaces and one is locally compact, the product space is a Kelley space. The product of any Kelley space is generally not a Kelley space. However , if you put it , a product is in the category of Kelley rooms.
- Hausdorff quotients of Kelley spaces are again Kelley spaces.
- One of the reasons why Kelley spaces are used in algebraic topology is the following statement: If and are Kelley spaces and denotes the space of continuous functions with the compact open topology , then the following evaluation mapping is continuous:
characterization
The following characterization of the Kelley spaces goes back to DE Cohen and shows that the Kelley spaces can be regarded as a generalization of the locally compact spaces:
- A Hausdorff space is a Kelley space if and only if it is the quotient of a locally compact space.
literature
- Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture (= BI university paperbacks . Volume 121 ). Bibliographisches Institut, Mannheim u. a. 1978, ISBN 3-411-00121-6 .
- James Dugundji : Topology (= Allyn and Bacon Series in Advanced Mathematics ). Allyn & Bacon, Boston MA 1966 (also: Brown, Dubuque IA 1989, ISBN 0-697-06889-7 ).
- Vasudevan Srinivas: Algebraic K-Theory (= Progress in Mathematics . Volume 90 ). 2nd ed. Birkhauser, Boston MA a. a. 1996, ISBN 0-8176-3702-8 .