In particular, under some of the definitions, they form a cartesian closed category while still containing the typical spaces of interest, which makes them convenient for use in algebraic topology.
There are multiple (non-equivalent) definitions of compactly generated space or k-space in the literature.
These definitions share a common structure, starting with a suitably specified family
is then compactly generated exactly when its topology is coherent with that family of subspaces; namely, a set
Another choice is to take the family of all continuous maps from arbitrary spaces of a certain type into
lead to different definitions of compactly generated space.
As an additional general note, a sufficient condition that can be useful to show that a space
is called compactly-generated or a k-space if it satisfies any of the following equivalent conditions:[2][3][4] As explained in the final topology article, condition (2) is well-defined, even though the family of continuous maps from arbitrary compact spaces is not a set but a proper class.
is called compactly-generated or a k-space if it satisfies any of the following equivalent conditions:[5][6][7] As explained in the final topology article, condition (1) is well-defined, even though the family of continuous maps from arbitrary compact Hausdorff spaces is not a set but a proper class.
Informally, a space whose topology is determined by its compact Hausdorff subspaces.
On the other hand, it satisfies Definition 2 because it is homeomorphic to the quotient space of the compact interval
Compactly generated spaces were originally called k-spaces, after the German word kompakt.
The motivation for their deeper study came in the 1960s from well known deficiencies of the usual category of topological spaces.
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.
[9] 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 nLab 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.
Another suggestion (1964) was to consider the usual Hausdorff spaces but use functions continuous on compact subsets.
These ideas generalize to the non-Hausdorff case;[10] i.e. with a different definition of compactly generated spaces.
[11] In modern-day algebraic topology, this property is most commonly coupled with the weak Hausdorff property, so that one works in the category CGWH of compactly generated weak Hausdorff spaces.
In order to express results in a more concise way, this section will make use of the abbreviations CG-1, CG-2, CG-3 to denote each of the three definitions unambiguously.
Such spaces can be called compactly generated Hausdorff without ambiguity.
removed is isomorphic to the Fortissimo space, which is not compactly generated (as mentioned in the Examples section, it is anticompact and non-discrete).
[16] Another example is the Arens space,[17][18] which is sequential Hausdorff, hence compactly generated.
For instance, as shown in the Examples section, there are many spaces that are not CG-1, but they are open in their one-point compactification, which is CG-1.
For a concrete example, the Sierpiński space is not CG-3, but is homeomorphic to the quotient of the compact interval
More generally, any final topology on a set induced by a family of functions from CG-1 spaces is also CG-1.
This follows by combining the results above for disjoint unions and quotient spaces, together with the behavior of final topologies under composition of functions.
with the quotient topology from the real line with the positive integers identified to a point is sequential.
Since compactly generated spaces are defined in terms of a final topology, one can express the continuity of