Compact-open topology

Definition
Suppose $$X$$ and $$Y$$ are topological spaces. The compact-open topology is a topology we can define on the space of continuous functions $$C(X,Y)$$ from $$X$$ to $$Y$$ as follows.

For a compact subset $$K \subseteq X$$ and an open subset $$U \subseteq Y$$, we define $$W(K,U)$$ as the set of all continuous maps $$f:X \to Y$$ such that $$f(K) \subseteq U$$. The compact-open topology is the topology with subbasis as the set of all $$W(K,U)$$s.