in topology, a compactly generated space (or k-space) is a topological space whose topology is coherent with the family of all compact subspaces. specifically, a topological space x is compactly generated if it satisfies the following condition: :a subspace a is closed in x if and only if a â© k is closed in k for all compact subspaces k â x. equivalently, one can replace closed with open in this definition. if x is coherent with any cover of compact subspaces in the above sense then it is, in fact, coherent with all compact subspaces. a compactly generated hausdorff space is a compactly generated space which is also hausdorff. like many compactness conditions, compactly generated spaces are often assumed to be hausdorff. ==motivation== compactly generated spaces were originally called k-spaces, after the german word kompakt. they were studied by hurewicz, and can be found in general topology by kelley, topology by dugundji, rational homotopy theory by fÃ©lix, halperin, thomas. the motivation for their deeper study came in the 1960s from well known deficiencies of the usual topological category. this fails to be a cartesian closed category, the usual cartesian product of identification maps is not always an identification map, and the usual product of cw-complexes need not be a cw-complex. by contrast, the category of simplicial sets had many convenient properties, including being cartesian closed. the history of the study of repairing this situation is given in the article on the ncatlab on convenient categories of spaces. the first suggestion (1962) to remedy this situation was to restrict oneself to the full subcategory of compactly generated hausdorff spaces, which is in fact cartesian closed. a definition of the exponential object is given below. another suggestion (1964) was to consider the usual hausdorff spaces but use functions continuous on compact subsets. these ideas can be generalised to the non-hausdorff case, see section 5.9 in the book topology and groupoids. this is useful since identification spaces of hausdorff spaces need not be hausdorff. for more information, see also the article by booth and tillotson. ==examples== most topological spaces commonly studied in mathematics are compactly generated. *every compact space is compactly generated. *every locally compact space is compactly generated. *every first-countable space is compactly generated. *topological manifolds are locally compact hausdorff and therefore compactly generated hausdorff. *metric spaces are first-countable and therefore compactly generated hausdorff. *every cw complex is compactly generated hausd
General_topology Homotopy_theory