The compact-open topology is the natural topology on spaces of continuous maps when one wants continuity to be tested uniformly on compact subsets.
Subbasic neighborhoods
A compact-open neighborhood prescribes where a map sends a compact set.
Definition. For spaces \(X,Y\), the compact-open topology on \(C(X,Y)\) has subbasis \[ [K,U]=\{f\in C(X,Y):f(K)\subseteq U\}, \] where \(K\subseteq X\) is compact and \(U\subseteq Y\) is open.
Proposition. If \(X\) is discrete, the compact-open topology on \(C(X,Y)=Y^X\) agrees with the product topology when compact subsets of \(X\) are finite.
Proof. For a discrete space in which compact sets are finite, subbasic compact-open conditions reduce to finitely many coordinate restrictions \(f(x)\in U_x\). These are exactly basic product conditions.
Uniform convergence on compact sets
For metric targets, compact-open convergence is uniform convergence on each compact subset under mild hypotheses.
Proposition. If \(Y\) is metric, a net \(f_\alpha\) in \(C(X,Y)\) converges to \(f\) in the compact-open topology precisely when for every compact \(K\subseteq X\) and every \(\varepsilon>0\), eventually \(\sup_{x\in K}d(f_\alpha(x),f(x))<\varepsilon\), provided \(X\) is locally compact Hausdorff.
Proof. Compact-open convergence gives eventual membership in neighborhoods of the form \([K,B(f(K),\varepsilon)]\). Compactness of \(K\) and continuity of \(f\) allow finitely many local target balls to be controlled uniformly. Conversely, uniform convergence on \(K\) directly implies eventual membership in every subbasic set \([K,U]\) containing \(f\), because compactness gives a positive distance buffer after refining \(U\) locally around \(f(K)\).
Exponential law
The compact-open topology is designed so that maps into function spaces correspond to maps on products.
Theorem. If \(X\) is locally compact Hausdorff, then a map \(F:Z\to C(X,Y)\) is continuous for the compact-open topology if and only if the adjoint map \(\widetilde F:Z\times X\to Y\), \(\widetilde F(z,x)=F(z)(x)\), is continuous.
Proof. If \(\widetilde F\) is continuous and \([K,U]\) is subbasic, then \(F^{-1}([K,U])\) consists of those \(z\) for which \(\widetilde F(\{z\}\times K)\subseteq U\). Local compactness of \(X\) lets one thicken compact neighborhoods around \(K\) and obtain an open condition on \(z\). Conversely, continuity of \(F\) makes evaluation continuous on compact neighborhoods: near \((z,x)\), choose a compact neighborhood \(K\) of \(x\) and a subbasic neighborhood \([K,U]\) of \(F(z)\). This gives a product neighborhood mapped into \(U\).
Evaluation map
Evaluation is not always continuous; local compactness of the source is the usual sufficient condition.
Corollary. If \(X\) is locally compact Hausdorff, the evaluation map \(\operatorname{ev}:C(X,Y)\times X\to Y\), \((f,x)\mapsto f(x)\), is continuous.
Proof. Apply the exponential law to the identity map \(C(X,Y)\to C(X,Y)\). Equivalently, if \(f(x)\in U\), choose a compact neighborhood \(K\) of \(x\) whose image under \(f\) lies in \(U\) after shrinking; then \([K,U]\times\operatorname{int}K\) is a neighborhood mapped into \(U\).
Remark. This is one reason locally compact Hausdorff spaces are especially natural as domains in function-space topology.
Exponential law and uniform convergence
Theorem (exponential law). If \(X\) is locally compact Hausdorff, then a map \[ F:Z\times X\to Y \] is continuous if and only if its adjoint \[ \widehat F:Z\to C(X,Y),\qquad \widehat F(z)(x)=F(z,x), \] is continuous when \(C(X,Y)\) has the compact-open topology.
The subbasic sets \[ [K,U]=\{f:f(K)\subset U\}, \qquad K\subset X\text{ compact},\ U\subset Y\text{ open}, \] are exactly what make evaluation and currying compatible. Local compactness provides a compact neighborhood on which continuity can be controlled uniformly.
Proposition. If \(X\) is compact and \(Y\) is metric, the compact-open topology on \(C(X,Y)\) is induced by \[ d_\infty(f,g)=\sup_{x\in X}d(f(x),g(x)). \]
Proof. A uniform ball controls the image of every compact subset and hence is compact-open. Conversely, from a finite intersection of subbasic neighborhoods of \(f\), compactness and the Lebesgue-number argument produce a positive uniform radius whose ball remains inside that intersection.