The compact-open topology on $C(X,Y)$ for a metric domain $X$, with subbasis $S(K,V) = \{f : f[K] \subseteq V\}$

Let $(X,d)$ be a metric space (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric) carrying its metric topology (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not), let $(Y, \mathcal{T}_Y)$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), and let

$C(X,Y) \;=\; \{\, f : X \to Y \;:\; f \text{ is continuous} \,\}$

(Continuity of a map of topological spaces at a point and globally). For a compact subset $K \subseteq X$ (Open cover, subcover, compact metric space, and compact subset of a metric space) and an open $V \in \mathcal{T}_Y$ put

$S(K,V) \;:=\; \{\, f \in C(X,Y) \;:\; f[K] \subseteq V \,\} \;=\; \{\, f \in C(X,Y) : f(x) \in V \text{ for every } x \in K \,\} .$

The compact-open topology on $C(X,Y)$ is the topology generated by

$\mathcal{S}_{\mathrm{co}} \;:=\; \{\, S(K,V) \;:\; K \subseteq X \text{ compact},\ V \in \mathcal{T}_Y \,\}$

as a subbasis (Basis and subbasis for a topology, and the topology generated by a family of sets). By A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis the finite intersections

$S(K_0,V_0) \cap \dots \cap S(K_{n-1},V_{n-1}) \qquad (n \in \mathbb{N})$

form a basis for it, the value $n = 0$ giving the empty intersection $C(X,Y)$. Nothing has to be checked for this to be a topology: a topology generated by an arbitrary family exists and is the coarsest one containing it (Basis and subbasis for a topology, and the topology generated by a family of sets).

Two degenerate members, recorded because they are used. $S(\varnothing, V) = C(X,Y)$ for every open $V$, the empty set being compact and $f[\varnothing] = \varnothing$; and $S(K, Y) = C(X,Y)$ for every compact $K$. Both are the whole space, so neither constrains anything, and arguments below dispose of them separately rather than dividing by a distance that does not exist.

The domain is metric, and the target is not. Compactness of $K$ is Open cover, subcover, compact metric space, and compact subset of a metric space, which is defined for subsets of a metric space and, at this point in the reading order, for nothing else; that is why $X$ carries a metric here. The target $Y$ needs only its open sets, so it is an arbitrary topological space throughout this definition and wherever the compact-open topology alone is in play. Where a distance in the target is used — the uniform metric, compact convergence, the comparison theorem — $Y$ is required to be metric and the requirement is stated.

Compactness of $K$ is intrinsic (Open cover, subcover, compact metric space, and compact subset of a metric space): it means that the metric subspace $(K, d_K)$ is a compact metric space. The equivalent description by families of open subsets of $X$ covering $K$ is A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, and it is cited at every step that uses it.

Notation. The letter $S$ carries two unrelated meanings in this library: $S(x,r)$ is the sphere of centre $x$ and radius $r$ in a metric space (Open ball, closed ball and sphere in a metric space), and $S(K,V)$ is the set defined above. The two are never ambiguous, because the first argument of a sphere is a point and its second a positive real, while the first argument of $S(K,V)$ is a compact set and its second an open set; no item on this page writes a sphere.

• Why compact sets and open sets. $S(K,V)$ says "$f$ maps all of $K$ into $V$". Taking $K$ to be a single point recovers the subbasis of the topology of pointwise convergence (The topology of pointwise convergence on $Y^{X}$, which is the product topology, and its restriction to $C(X,Y)$), so the compact-open topology is at least as fine as that one; taking $K$ large makes the condition a uniform one over $K$, which is what the comparison with the topology of compact convergence on this page makes precise.

• The definition is on $C(X,Y)$ and not on $Y^{X}$. The sets $S(K,V)$ could be written down for arbitrary functions, but the theory of this page uses that $f[K]$ is compact when $K$ is and $f$ is continuous (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset), which is false for a discontinuous $f$. The compact-open topology in this library is therefore a topology on continuous maps only.

• Metrizability is not asserted. The compact-open topology need not be metrizable, and this page records that as a false statement with an explicit witness. What is proved here is that for a metric target it coincides with the topology of compact convergence, which for a suitable $X$ is metrizable by a metric this library does not construct.