Compactly Generated Space - Properties

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 {K_α} 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 ∩ K_α is closed in K_α for each α. Denote this new space by X_c. One can show that the compact subsets of X_c and X coincide and the induced topologies are the same. It follows that X_c is compactly generated. If X was compactly generated to start with then X_c = X otherwise the topology on X_c 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 X_c 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.

These ideas can be generalised to the non-Hausdorff case, see section 5.9 in the book `Topology and groupoids' listed below. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.