When in a convenient category of topological spaces, e.g. compactly generated spaces, the category is cartesian closed, so that there is an adjunction between the mapping space and the cartesian product in that category. For general topological spaces there is no globally defined adjunction, but we can instead characterize exactly which spaces are exponentiable.
This condition, however, is not really any more explicit. More interesting is to characterize the exponentiable spaces in terms of a point-set-topological condition.
For open subsets and of a topological space , we write to mean that any open cover of admits a finite subcover of ; this is read as is relatively compact under or is way below . We say that is core-compact if for every open neighborhood of a point , there exists an open neighborhood of with . In other words, is core-compact iff for all open subsets , we have . This says essentially the same thing as saying that the open-set lattice of is a continuous lattice, which yields the corresponding definition for locales.
An object of is exponentiable if and only if it is core-compact.
If is Hausdorff, then core-compactness is equivalent to local compactness; thus in particular all locally compact Hausdorff spaces are exponentiable. In fact, local compactness implies exponentiability even without the Hausdorff condition, if local compactness is defined appropriately (for every point the compact neighborhoods form a neighborhood basis). For this reason, that core-compactness is also called quasi local compactness.
When is core-compact, we can explicitly describe the exponential topology on (whose points are continuous maps ). It is generated by subbasis elements , for an open subset of and an open subset of , where a continuous map belongs to iff :
If and are Hausdorff, then this topology on coincides with the compact-open topology.
Exponentiable (i.e. core-compact) spaces can also be characterized in terms of ultrafilter convergence. Recall that a topological space can equivalently be defined as a lax algebra for the ultrafilter monad on the (1,2)-category Rel of sets and relations. In other words, it consists of a set and a relation called “convergence”, such that and , where and are the unit and multiplication of the ultrafilter monad, regarded as relations. In the paper
it is shown that a space is exponentiable (i.e. core-compact) if and only if we have equality in the multiplication law .
Some intuition for this characterization can be obtained as follows. Consider the standard non-locally-compact space, the rationals as a subspace of the reals . Suppose that is a rational number and that is a sequence of irrationals converging to . Then for each we can find a sequence of rationals which converges to ; hence the form a “sequence of sequences” which “globally converges” to in , i.e. which are related to by the composite relation , but for which does not converge elementwise to an intermediate sequence which in turn converges to , i.e. it is not related to by the relation . It turns out that when generalized to ultrafilter convergence, this sort of behavior exactly characterizes what it means to fail to be (quasi) locally compact.
If is exponentiable, then the exponential law gives us an isomorphism of sets for any other spaces and . If is also exponentiable, then the Yoneda lemma yields from this a homeomorphism . However, we can also say some things in general without all spaces involved being exponentiable.
We now agree to denote by the space of continuous maps in the compact-open topology.
Let be topological spaces. For any , the formula
The adjunction map
If both assumptions (on and ) are satisfied, then is not only a continuous bijection, but also open, hence a homeomorphism.
There is also a version for based (= pointed) topological spaces. The cartesian product then needs to be replace by the smash product of the based spaces. Regarding that the maps preserve the base point, the adjunction map induces the adjunction map
where the mapping space for based spaces is the subspace of the usual mapping space, in the compact-open topology, which consists of the mappings preserving the base point.
It appears that is again one-to-one and continuous, and it is bijective if is locally compact Hausdorff. If is also Hausdorff then is a homeomorphism.
M. Escardo and R. Heckmann, Topologies of spaces of continuous functions, 2001.
Eva Lowen-Colebunders and Günther Richter, An elementary approach to exponential spaces, MR.