# Topology on function Spaces

I was wondering, what are the different topologies that are usually given to function spaces? Sorry if this is a broad question.

For example, I know that compact-open (under nice assumptions) leads to $\mathcal{C}(X\times A,Y)\cong \mathcal{C}(X,Y^A)$. What are other typical useful topologies given to function spaces? Why?

• This is the field of functional analysis. It's pretty big. Dec 3, 2013 at 5:05
• Two important ones are the topology of pointwise convergence and the topology or uniform convergence (for metric, pseudometric, and uniform spaces). Beyond that, the question is too broad. Dec 3, 2013 at 5:30
• There is the test-open topology, though it should maybe be called compact-Hausdorff-open topology. In that case we define a test map to be a continuous map from a compact Hausdorff space $C$ to $X$, and $K(X,Y)$ to be the set of all $k$-maps, that are functions $f:X→Y$ such that $ft$ is continuous for all test maps $t$. The topology has as subbase the sets $W(t,U)=\{f\mid ft[C]\subseteq U\}$ with $t$ test map and $U$ open. We can prove that $K(X×Y,Z)≅K(X,K(Y,Z))$. If we consider the case where $X×Y$ is a $k$-space (every $k$-map is a map), then this formula becomes $Z^{X×Y}≅(Z^Y)^X$ Dec 3, 2013 at 14:52
• What about the product topology? Jul 15, 2021 at 18:47