Compactly generated space
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In topology, a compactly generated space is a topological space X satisfying the following condition: a subspace A is closed in X if and only if A ∩ K is closed in K for all compact subspaces K ⊆ X. Equivalently, one can replace closed with open in this definition.
Most topological spaces commonly studied in mathematics are compactly generated. For instance, every locally compact (and compact) space is compactly generated, as is every first-countable space. Additionally, every CW complex is compactly generated.
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, i.e. compactly generated spaces which are also Hausdorff. This category is, in fact, cartesian closed. A definition of the exponential object is given below. The category of compactly generated Hausdorff spaces is general enough to include all metric spaces, topological manifolds, and all CW complexes.
[edit] Properties
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 CGHaus which takes X to Xc is left adjoint to the forgetful functor CGHaus → 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.
