A topology on the space C(X,Y) of continuous maps from X to Y generated by subbasic sets of the form [K,U]={f∈C(X,Y):f(K)⊂U}, where K⊂X is compact and U⊂Y is open; it encodes control of maps on compact subsets of the domain.

Principle

Use compact subsets of the domain together with open targets to produce neighbourhoods of functions; convergence means uniform control on each fixed compact set rather than on the whole domain or merely at points.

Demonstration

If X is locally compact and Hausdorff and Y is a metric space, the compact-open topology on C(X,Y) coincides with the topology of uniform convergence on compact subsets: a sequence f_n→f iff sup_{x∈K} d(f_n(x),f(x))→0 for every compact K⊂X.

Misapplication

Assuming compact-open topology implies uniform convergence on the whole domain when X is noncompact or believing it is always metrizable can lead to false statements about continuity of limits or compactness of function families.

Consequence

Evaluation maps ev_x:C(X,Y)→Y are continuous for each x∈X and composition maps are often continuous under mild hypotheses; the topology is well-suited to homotopy theory and mapping-space constructions because it respects compact control.

Reversal

Pointwise (product) topology is weaker — it only controls values at individual points — while the topology of uniform convergence on the whole domain is stronger; compact-open sits between these extremes when X is noncompact.

Boundary

Defined for continuous maps between topological spaces and depends on the compact subsets of X; it is most useful when X has many compact sets (locally compact spaces); it does not automatically give metric or completeness properties of C(X,Y).

Semantic Tension

Competes conceptually with the product (pointwise) topology and with uniform convergence topologies; in some contexts the compact-open topology aligns with the topology induced by the uniform metric on compacta, creating ambiguity about which structure is fundamental.

Synthesis

The compact-open topology provides a function-space topology that measures closeness by requiring uniform control on compact subsets: neighborhoods specify that the image of a compact set lie inside a chosen open target, blending compactness and continuity into convergence conditions.