In topology, a topological space is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other. Also some authors include some separation axiom (like Hausdorff space or weak Hausdorff space) in the definition of one or both terms, and others do not.
In the simplest definition, a compactly generated space is a space that is coherent with the family of its compact subspaces, meaning that for every set is open in if and only if is open in for every compact subspace Other definitions use a family of continuous maps from compact spaces to and declare to be compactly generated if its topology coincides with the final topology with respect to this family of maps. And other variations of the definition replace compact spaces with compact Hausdorff spaces.
Compactly generated spaces were developed to remedy some of the shortcomings of the category of topological spaces. 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. For more details on this, see the category of compactly generated weak Hausdorff spaces.
Definitions
General framework for the definitions
Let be a topological space, where is the topology, that is, the collection of all open sets in
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 of continuous maps from some compact spaces to The various definitions differ in their choice of the family as detailed below.
The final topology on with respect to the family is called the k-ification of Since all the functions in were continuous into the k-ification of is finer than (or equal to) the original topology . The open sets in the k-ification are called the in they are the sets such that is open in for every in Similarly, the in are the closed sets in its k-ification, with a corresponding characterization. In the space every open set is k-open and every closed set is k-closed. The space together with the new topology is usually denoted
The space is called compactly generated or a k-space (with respect to the family ) if its topology is determined by all maps in , in the sense that the topology on is equal to its k-ification; equivalently, if every k-open set is open in or if every k-closed set is closed in or in short, if
As for the different choices for the family , one can take all the inclusions maps from certain subspaces of for example all compact subspaces, or all compact Hausdorff subspaces. This corresponds to choosing a set of subspaces of The space is then compactly generated exactly when its topology is coherent with that family of subspaces; namely, a set is open (resp. closed) in exactly when the intersection is open (resp. closed) in for every Another choice is to take the family of all continuous maps from arbitrary spaces of a certain type into for example all such maps from arbitrary compact spaces, or from arbitrary compact Hausdorff spaces.
These different choices for the family of continuous maps into lead to different definitions of compactly generated space. Additionally, some authors require to satisfy a separation axiom (like Hausdorff or weak Hausdorff) as part of the definition, while others don't. The definitions in this article will not comprise any such separation axiom.
As an additional general note, a sufficient condition that can be useful to show that a space is compactly generated (with respect to ) is to find a subfamily such that is compactly generated with respect to For coherent spaces, that corresponds to showing that the space is coherent with a subfamily of the family of subspaces. For example, this provides one way to show that locally compact spaces are compactly generated.
Below are some of the more commonly used definitions in more detail, in increasing order of specificity.
For Hausdorff spaces, all three definitions are equivalent. So the terminology is unambiguous and refers to a compactly generated space (in any of the definitions) that is also Hausdorff.
Definition 1
Informally, a space whose topology is determined by its compact subspaces, or equivalently in this case, by all continuous maps from arbitrary compact spaces.
A topological space is called compactly-generated or a k-space if it satisfies any of the following equivalent conditions:
(1) The topology on is coherent with the family of its compact subspaces; namely, it satisfies the property: a set is open (resp. closed) in exactly when the intersection is open (resp. closed) in for every compact subspace (2) The topology on coincides with the final topology with respect to the family of all continuous maps from all compact spaces (3) is a quotient space of a topological sum of compact spaces. (4) is a quotient space of a weakly locally compact space.
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.
The equivalence between conditions (1) and (2) follows from the fact that every inclusion from a subspace is a continuous map; and on the other hand, every continuous map from a compact space has a compact image and thus factors through the inclusion of the compact subspace into
Definition 2
Informally, a space whose topology is determined by all continuous maps from arbitrary compact Hausdorff spaces.
A topological space is called compactly-generated or a k-space if it satisfies any of the following equivalent conditions:
(1) The topology on coincides with the final topology with respect to the family of all continuous maps from all compact Hausdorff spaces In other words, it satisfies the condition: a set is open (resp. closed) in exactly when is open (resp. closed) in for every compact Hausdorff space and every continuous map (2) is a quotient space of a topological sum of compact Hausdorff spaces. (3) is a quotient space of a locally compact Hausdorff space.
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.
Every space satisfying Definition 2 also satisfies Definition 1. The converse is not true. For example, the one-point compactification of the Arens-Fort space is compact and hence satisfies Definition 1, but it does not satisfies Definition 2.
Definition 2 is the one more commonly used in algebraic topology. This definition is often paired with the weak Hausdorff property to form the category CGWH of compactly generated weak Hausdorff spaces.
Definition 3
Informally, a space whose topology is determined by its compact Hausdorff subspaces.
A topological space is called compactly-generated or a k-space if its topology is coherent with the family of its compact Hausdorff subspaces; namely, it satisfies the property: a set is open (resp. closed) in exactly when the intersection is open (resp. closed) in for every compact Hausdorff subspace
Every space satisfying Definition 3 also satisfies Definition 2. The converse is not true. For example, the Sierpiński space with topology does not satisfy Definition 3, because its compact Hausdorff subspaces are the singletons and , and the coherent topology they induce would be the discrete topology instead. On the other hand, it satisfies Definition 2 because it is homeomorphic to the quotient space of the compact interval obtained by identifying all the points in
By itself, Definition 3 is not quite as useful as the other two definitions as it lacks some of the properties implied by the others. For example, every quotient space of a space satisfying Definition 1 or Definition 2 is a space of the same kind. But that does not hold for Definition 3.
However, for weak Hausdorff spaces Definitions 2 and 3 are equivalent. Thus the category CGWH can also be defined by pairing the weak Hausdorff property with Definition 3, which may be easier to state and work with than Definition 2.
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 Félix, Halperin, and Thomas.
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. (See the Appendix) 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. These ideas extend on the de Vries duality theorem. 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 generalize to the non-Hausdorff case; i.e. with a different definition of compactly generated spaces. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.
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.
Examples
As explained in the Definitions section, there is no universally accepted definition in the literature for compactly generated spaces; but Definitions 1, 2, 3 from that section are some of the more commonly used. 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. This is summarized in the table below (see the Definitions section for other equivalent conditions for each).
Abbreviation Meaning summary CG-1 Topology coherent with family of its compact subspaces CG-2 Topology same as final topology with respect to continuous maps from arbitrary compact Hausdorff spaces CG-3 Topology coherent with family of its compact Hausdorff subspaces
For Hausdorff spaces the properties CG-1, CG-2, CG-3 are equivalent. Such spaces can be called compactly generated Hausdorff without ambiguity.
Every CG-3 space is CG-2 and every CG-2 space is CG-1. The converse implications do not hold in general, as shown by some of the examples below.
For weak Hausdorff spaces the properties CG-2 and CG-3 are equivalent.
Sequential spaces are CG-2. This includes first countable spaces, Alexandrov-discrete spaces, finite spaces.
Every CG-3 space is a T1 space (because given a singleton its intersection with every compact Hausdorff subspace is the empty set or a single point, which is closed in hence the singleton is closed in ). Finite T1 spaces have the discrete topology. So among the finite spaces, which are all CG-2, the CG-3 spaces are the ones with the discrete topology. Any finite non-discrete space, like the Sierpiński space, is an example of CG-2 space that is not CG-3.
Compact spaces and weakly locally compact spaces are CG-1, but not necessarily CG-2 (see examples below).