then its theorems often require an additional assumption of Hausdorffness somewhere. Every Hausdorff-compactly generated space is compactly generated. Every (Hausdorff) compact space is (Hausdorff-)compactly generated. Every weakly locally compact space is compactly generated. Every locally compact Hausdorff space is Hausdorff-compactly generated. Every first-countable (Hausdorff) space is (Hausdorff-)compactly generated. Topological manifolds are locally compact Hausdorff and therefore Hausdorff-compactly generated. Metric spaces are first-countable Hausdorff and therefore Hausdorff-compactly generated Hausdorff. Every CW complex is Hausdorff-compactly generated and Hausdorff. Every sequential space is a quotient of a metric space and therefore Hausdorff-compactly generated. Examples of topological spaces that fail to be compactly generated include the following: The product space endowed with the product topology, where the first factor uses the subspace topology while the second factor is the quotient space of where all natural numbers are identified with a single point. If is a non-principal ultrafilter on an infinite set the induced topology has the property that every compact set is finite, and is not compactly generated. A subspace of a compactly generated space is not necessarily compact generated. Although is compactly generated, the product of uncountably many copies of is not compactly generated. A Hausdorff space is compactly generated if and only if it is a quotient of some locally compact space (specifically, it is a quotient of all disjoint union of all compact subspace of ). Properties In a compactly generated space, every closed set is compactly generated. The same does not hold for open sets. For example, if denotes the Arens-Fort space, the one-point compactification of is compact, hence compactly generated, and it contains as an open subspace. But is not compactly generated (because it is Hausdorff and all its compact subsets are finite; so the k-ification of is the discrete topology, which is not the original topology on ). In a Hausdorff-compactly generated space, every closed set and every open set is Hausdorff-compactly generated. The same holds more generally for every locally closed set, that is, the intersection of an open set and a closed set. A quotient of a (Hausdorff)-compactly generated space is (Hausdorff)-compactly generated. A disjoint union of (Hausdorff)-compactly generated spaces is (Hausdorff)-compactly generated. A wedge sum of (Hausdorff)-compactly generated spaces is (Hausdorff)-compactly generated. The continuity of a map defined on a (Hausdorff-)compactly generated space can be determined solely by looking at the compact (Hausdorff) subsets of Specifically, a function on a (Hausdorff-)compactly generated space is continuous