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 k-open sets in they are the sets such that is open in for every in Similarly, the k-closed sets 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 1
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 compactly generated Hausdorff space 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:234
(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
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:567
(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
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.5
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.8 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.
Properties
(See the Examples section for the meaning of the abbreviations CG-1, CG-2, CG-3.)
Subspaces
Subspaces of a compactly generated space are not compactly generated in general, even in the Hausdorff case. For example, the ordinal space where is the first uncountable ordinal is compact Hausdorff, hence compactly generated. Its subspace with all limit ordinals except 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,1718 which is sequential Hausdorff, hence compactly generated. It contains as a subspace the Arens-Fort space, which is not compactly generated.
In a CG-1 space, every closed set is CG-1. The same does not hold for open sets. 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.
In a CG-2 space every closed set is CG-2; and so is every open set (because there is a quotient map for some locally compact Hausdorff space and for an open set the restriction of to is also a quotient map on a locally compact Hausdorff space). The same is true more generally for every locally closed set, that is, the intersection of an open set and a closed set.19
In a CG-3 space, every closed set is CG-3.
Quotients
The of a family of topological spaces is CG-1 if and only if each space is CG-1. The corresponding statements also hold for CG-2 and CG-3.
