Compactly generated space
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In topology, a compactly generated space (or k-space) is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition:
Equivalently, one can replace closed with open in this definition. If X is coherent with any cover of compact subsets in the above sense then it is, in fact, coherent with all compact subsets.
A compactly generated Hausdorff space is a compactly generated space which is also Hausdorff. Like many compactness conditions, compactly generated spaces are often assumed to be Hausdorff.
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[edit] Motivation
One of the primary motivations for studying compactly generated spaces comes from category theory. The category of topological spaces, Top, is defective in the sense that it fails to be a cartesian closed category. There have been various attempts to remedy this situation, one of which is to restrict oneself to the full subcategory of compactly generated Hausdorff spaces. This category is, in fact, cartesian closed. A definition of the exponential object is given below.
[edit] Examples
Most topological spaces commonly studied in mathematics are compactly generated.
- Every compact space is compactly generated.
- Every locally compact space is compactly generated.
- Every first-countable space is compactly generated.
- Topological manifolds are locally compact Hausdorff and therefore compactly generated Hausdorff.
- Metric spaces are first-countable and therefore compactly generated Hausdorff.
- Every CW complex is compactly generated Hausdorff.
[edit] Properties
We denote CGTop the full subcategory of Top with objects the compactly generated spaces, and CGHaus the full subcategory of CGTop with objects the Hausdorff separated spaces.
Given any topological space X we can define a (possibly) finer topology on X which is compactly generated. Let {Kα} denote the family of compact subsets of X. We define the new topology on X by declaring a subset A to be closed if and only if A ∩ Kα is closed in Kα for each α. Denote this new space by Xc. One can show that the compact subsets of Xc and X coincide and the induced topologies are the same. It follows that Xc is compactly generated. If X was compactly generated to start with then Xc = X otherwise the topology on Xc is strictly finer than X (i.e. there are more open sets).
This construction is functorial. The functor from Top to CGTop which takes X to Xc is right adjoint to the inclusion functor CGTop → Top.
The continuity of a map defined on compactly generated space X can be determined solely by looking at the compact subsets of X. Specifically, a function f : X → Y is continuous if and only if it is continuous when restricted to each compact subset K ⊆ X.
If X and Y are two compactly generated spaces the product X × Y may not be compactly generated (it will be if at least one of the factors is locally compact). Therefore when working in categories of compactly generated spaces it is necessary to define the product as (X × Y)c.
The exponential object in the CGHaus is given by (YX)c where YX is the space of continuous maps from X to Y with the compact-open topology.
[edit] See also
[edit] References
- Mac Lane, Saunders (1998). Categories for the Working Mathematician, (2nd ed.), Graduate Texts in Mathematics 5, Springer-Verlag. ISBN 0-387-98403-8.
- Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6 (Dover edition).
