Math 527 - Homotopy Theory Additional notes

The collection of k-closed subsets of X forms a topology, which contains the original topology of X (i.e. closed subsets are always k-closed). Notation 1.2. Let kX denote the space whose underlying set is that of X, but equipped with the topology of k-closed subsets of X. Because the k-topology contains the original topology on X, the identity function id : kX → X is continuous. Definition 1.3. A space X is compactly generated (CG), sometimes called a k-space, if kX → X is a homeomorphism. In other words, every k-closed subset of X is closed in X. Example 1.4. Every locally compact space is CG. Example 1.5. Every first-countable space is CG. More generally, every sequential space is CG. Example 1.6. Every CW-complex is CG. Notation 1.7. Let CG denote the full subcategory of Top consisting of compactly generated spaces. Notation 1.8. The construction of kX defines a functor k : Top→ CG, called the k-ification functor. Proposition 1.9. Let X be a CG space and Y an arbitrary space. Then a function f : X → Y is continuous if and only if for every compact Hausdorff space K and continuous map u : K → X, the composite fu : K → Y is continuous.