The exponential law: for a locally compact metric $X$ and any spaces $Z$ and $Y$, transposition is a bijection between $C(X \times Z, Y)$ and $C(Z, C(X,Y))$ with the compact-open topology

Let $(X,d)$ be a locally compact metric space (Locally compact metric space: every point has a compact neighbourhood) carrying its metric topology, and let $Z$ and $Y$ be topological spaces (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). Give $C(X,Y)$ the compact-open topology (The compact-open topology on $C(X,Y)$ for a metric domain $X$, with subbasis $S(K,V) = \{f : f[K] \subseteq V\}$) and $X \times Z$ the product topology (The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space). Define, for $f \in C(X \times Z, Y)$,

$\Phi(f) : Z \to C(X,Y), \qquad \Phi(f)(z)(x) := f(x,z) .$

Then $\Phi$ is a well-defined map $C(X \times Z, Y) \to C\big(Z, C(X,Y)\big)$ and it is a bijection (Injection, surjection, bijection); its inverse sends a continuous $F : Z \to C(X,Y)$ to the continuous map $(x,z) \mapsto F(z)(x)$.

Exactly what is and is not claimed

This is an assertion about two sets of continuous maps and a bijection between them. No topology is placed on $C(X \times Z, Y)$ or on $C(Z, C(X,Y))$ anywhere in the statement, and it is not claimed that $\Phi$ is a homeomorphism. The homeomorphism form of the exponential law is a genuinely stronger statement, and this library does not have what it needs; the last remark below says exactly what is missing. A reader who wants the categorical slogan "$Y^{X \times Z} \cong (Y^{X})^{Z}$" should read it here as a bijection of underlying sets, natural in the evident way, and no more.

No choice principle is used.

• Where each hypothesis is spent. Continuity of $\Phi(f)$, which is injectivity's half of the correspondence, needs nothing beyond compactness being available for subsets of $X$ (If $f : X \times Z \to Y$ is continuous then its transpose $F : Z \to C(X,Y)$, $F(z)(x) = f(x,z)$, is continuous for the compact-open topology, with no hypothesis on $X$ beyond being metric). Surjectivity is where local compactness enters, and it enters once, through continuity of the evaluation map at step 3.1 (If $X$ is a locally compact metric space then the evaluation map is continuous for the compact-open topology). Without it the map $\Psi(F)$ need not be continuous and $\Phi$ need not be onto.

• What the homeomorphism form would additionally need, stated precisely. To say that $\Phi$ is a homeomorphism one must first topologise both sides, which means giving $C(X \times Z, Y)$ a compact-open topology built over the compact subsets of $X \times Z$, and $C(Z, C(X,Y))$ one built over the compact subsets of $Z$. Neither is available here, and not for want of a notion of compactness: compactness for an arbitrary topological space (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right) and the tube lemma for a compact factor of an arbitrary product (Tube lemma: if $K$ is compact and an open $N \subseteq X \times Z$ contains $K \times \{z_0\}$, then $N$ contains $K \times W$ for some open $W \ni z_0$) are both developed earlier in the reading order, so "compact subset of $Z$" and "compact subset of $X \times Z$" do have meaning. What is missing is the topology itself: The compact-open topology on $C(X,Y)$ for a metric domain $X$, with subbasis $S(K,V) = \{f : f[K] \subseteq V\}$ is stated for a metric domain, whereas $Z$ here is an arbitrary topological space and $X \times Z$ carries no metric, so neither side is topologised by anything on this page. Beyond that, the direction that $\Phi^{-1}$ is continuous also needs that a compact subset of $X \times Z$ be covered by finitely many products of compacta, which this library does not prove. None of that is done here, and nothing above assumes it.

• The bijection is what the name usually denotes. Most treatments state the exponential law first as this correspondence and only afterwards ask when it is a homeomorphism, the answer requiring hypotheses on $Z$ as well as on $X$. The scope taken here is therefore the standard first form, and it is stated as such rather than as a weakened version of something else.

• The companion page traces the correspondence through an explicit example, the multiplication map on $\mathbb{R} \times \mathbb{R}$ and its transpose, and checks both halves by hand.