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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?

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  • $\begingroup$ This is the field of functional analysis. It's pretty big. $\endgroup$ Dec 3, 2013 at 5:05
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    $\begingroup$ 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. $\endgroup$
    – dfeuer
    Dec 3, 2013 at 5:30
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    $\begingroup$ 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 $Z^{X×Y}≅(Z^Y)^X$ $\endgroup$ Dec 3, 2013 at 14:52
  • $\begingroup$ What about the product topology? $\endgroup$
    – John M-D94
    Jul 15, 2021 at 18:47

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Since function spaces C(X,Y) itself makes a topological space so there are many ways of topolizing it. The most used topologies are Topology of pointwise convergence, Compact open topology, and Topology of uniform convergence. There are newly new topologies defined on C(X,Y) like Fine topology, Graph topology, Regular topology. Moreover topologists are generalizing this concept while taking the function space as C(X) = C(X,R), set of all continuous functions on X, where X being Tychonoff space. There is a good number of topologies defined on this space which also have comparable relations with the above mentioned topologies. Topologies on Function spaces is itself a broader area of research.

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