Compact-open topology

This article defines a function space topology i.e. a topology on the collection of continuous maps between two topological spaces

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