nLab
exponential law for spaces

When in a convenient category of topological spaces, e.g. compactly generated spaces, there is an adjunction between the mapping space and the product in that category. For general topological spaces there is no globally defined adjunction, instead we need some conditions. Here is the statement containing all the subtleties.

For $C$ a category with finite products, recall that an object $c$ is exponentiable if the functor $c\times -:C\to C$ has a right adjoint, usually denoted $(-{)}^c:C\to C$.

Theorem (Exponentiability)

Let Top be the category of all topological spaces. An object $X$ of $\mathrm{Top}$ is exponentiable if and only if $X$ satisfies either of these equivalent conditions:

• $X\times -:\mathrm{Top}\to \mathrm{Top}$ preserves coequalizers. Notice that $X\times -:\mathrm{Top}\to \mathrm{Top}$ always preserves coproducts, so this condition is the same as that $X\times -$ preserves colimits. This is equivalent to exponentiability by the adjoint functor theorem.

• $X$ is core-compact: for every open neighborhood $U$ of a point $x$, there exists an open neighborhood $V$ of $x$ such that for every open cover of $U$, there exists a finite subcover of $V$. This is a theorem of hard topology, not abstract nonsense like the previous one.

If $X$ is Hausdorff, then core-compactness is equivalent to local compactness.

If $X$ is core-compact, then the topology on $Y^X$ (whose points are identified with continuous maps $f:X\to Y$) is generated by subbasis? elements $O_{U,V}$, for $U$ an open subset of $X$ and $V$ an open subset of $Y$; a continuous map $f:X\to Y$ belongs to $O_{U,V}$ iff every open cover of $f^{-1}(V)$ admits a finite subcover of $U$:

$$O_{U,V}=\{f\in Y^X:\mathrm{every}\mathrm{open}\mathrm{cover}\mathrm{of}{f}^{-1}(V)\mathrm{admits}a\mathrm{finite}\mathrm{subcover}\mathrm{of}U\}.$$O_{U, V} = \{f \in Y^X: every open cover of f^{-1}(V) admits a finite subcover of U\} .

If $X$ and $Y$ are Hausdorff, then the topology on $Y^X$ coincides with the compact-open topology.

Further remarks should be made which connect this to the theory of continuous lattice?s (essentially, a space is exponentiable if its topology is a continuous lattice).

Denote by $\mathrm{Map}(X,Y)=X^Y$ the space of continuous maps $X\to Y$ in the compact-open topology.

Theorem (Exponential law)

Let $X,Y,B$ be topological spaces. For any $f\in B^{X\times Y}$, the formula

$$[(\theta f)(y)](x)=f(x,y)$$[(\theta f)(y)](x) = f(x,y)

defines a continuous map $\theta f:Y\to B^X$ which we call the map adjoint to $f$, or the adjunct of $f$.

The adjunction map

$$\theta :\mathrm{Map}(X\times Y,B)\to \mathrm{Map}(Y,B^X),\theta :f\mapsto \theta f$$\theta : Map(X\times Y,B)\to Map(Y,B^X), \,\,\,\,\,\,\,\theta:f\mapsto \theta f

is a one-to-one function, and if $X$ is locally compact and Hausdorff then $\theta$ is a bijection. Independently from that assumption on $X$, if $Y$ is Hausdorff, then $\theta$ is continuous in the compact-open topology

$$\theta :B^{X\times Y}\to (B^X{)}^Y.$$\theta : B^{X\times Y}\to (B^X)^Y.

If both assumptions (on $X$ and $Y$) are satisfied, then $\theta$ 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 $\theta$ induces the adjunction map

$${\theta }_{*}:{\mathrm{Map}}_{*}(X\wedge Y,B)\to {\mathrm{Map}}_{*}(Y,B^X)$$\theta_*:Map_*(X\wedge Y,B)\to Map_*(Y,B^X)

where the mapping space ${\mathrm{Map}}_{*}$ 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 ${\theta }_{*}$ is again one-to-one and continuous, and it is bijective if $X$ is locally compact Hausdorff. If $Y$ is also Hausdorff then ${\theta }_{*}$ is a homeomorphism.

Reference

M. Escardo and R. Heckmann, Topologies of spaces of continuous functions, 2001.