if and only if the same is true of its restriction to each compact (Hausdorff) subset If is (Hausdorff-)compactly generated and is locally compact (Hausdorff), then the product is (Hausdorff-)compactly generated. If and are two (Hausdorff-)compactly generated spaces, then may not be (Hausdorff-)compactly generated. Therefore, when working in categories of (Hausdorff-)compactly generated spaces it is necessary to define the product as the k-ification of the product topology (see below). If is compactly generated and the space of continuous functions has the compact-open topology then the path components in are precisely the homotopy equivalence classes. K-ification Given any topological space we can define a possibly finer topology on that is compactly generated, sometimes called the of the topology. Let denote the family of compact subsets of We define the new topology on by declaring a subset to be closed if and only if is closed in for each index Denote this new space by One can show that the compact subsets of and coincide, and the induced topologies on compact subsets are the same. It follows that is compactly generated. If was compactly generated to start with then Otherwise the topology on is strictly finer than (i.e. there are more open sets). This construction is functorial. We denote the full subcategory of with objects the compactly generated spaces, and the full subcategory of with objects the Hausdorff spaces. The functor from to that takes to is right adjoint to the inclusion functor The above discussion applies also to the Hausdorff-compactly generated spaces after replacing compact with compact Hausdorff, but with the following difference. To prove that the compact Hausdorff subsets of the k-ification are the same as in the original topology (and hence that the k-ification is Hausdorff-compactly generated) requires that the original topology is also k-Hausdorff. The following properties are equivalent: Hausdorff-compactly generated k-Hausdorff Hausdorff-compactly generated weak Hausdorff Compactly generated k-Hausdorff The exponential object in is given by where is the space of continuous maps from to with the compact-open topology. These ideas can be generalized to the non-Hausdorff case. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff. See also Notes References J. Peter May, A Concise Course in Algebraic Topology, (1999) Chicago Lectures in Mathematics (See Chapter 5.) External links Compactly generated spaces - contains an excellent catalog of properties and constructions with compactly generated spaces General topology Homotopy theory