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: ${\displaystyle X}$

• ${\displaystyle X}$is a Hausdorff room
• A subset is closed if and only if the averages for all compact subsets is closed. ${\displaystyle A\subset X}$ ${\displaystyle A\cap K}$ ${\displaystyle K\subset X}$

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. ${\displaystyle (X,\tau )}$ ${\displaystyle \tau _{K}}$ ${\displaystyle U\in \tau _{K}:\Leftrightarrow (X\setminus U)\cap K}$ ${\displaystyle K\subset X}$ ${\displaystyle \tau _{K}}$ ${\displaystyle X}$ ${\displaystyle \tau \subset \tau _{K}}$ ${\displaystyle X}$ ${\displaystyle (X,\tau _{K})}$ ${\displaystyle X}$ ${\displaystyle k(X)}$

${\displaystyle (X,\tau )}$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. ${\displaystyle \tau =\tau _{K}}$ ${\displaystyle \tau _{K}}$ ${\displaystyle X}$

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 . ${\displaystyle f:X\rightarrow Y}$ ${\displaystyle f:k(X)\rightarrow k(Y)}$ ${\displaystyle k:{\mathcal {H}}\rightarrow {\mathcal {K}}}$ ${\displaystyle I:{\mathcal {K}}\rightarrow {\mathcal {H}}}$ ${\displaystyle I}$ ${\displaystyle k}$

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 .) ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f:X\rightarrow Y}$ ${\displaystyle f}$ ${\displaystyle \Leftrightarrow f|_{K}}$ ${\displaystyle K\subset X}$ ${\displaystyle {\rm {id}}_{X}:X\rightarrow k(X)}$
• 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. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X\times Y}$ ${\displaystyle X\times _{k}Y:=k(X\times Y)}$ ${\displaystyle \times _{k}}$
• 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: ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle C_{co}(X,Y)}$ ${\displaystyle X\to Y}$

${\displaystyle k(C_{co}(X,Y))\times _{k}X\rightarrow Y,\,\,(f,x)\mapsto f(x)}$

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 .