  Many classically used function space structures (including the topology of
pointwise convergence, the compact-open topology, the Isbell topology and the
continuous convergence) are induced by a hyperspace structure counterpart. This
scheme is used to study local properties of function space structures on
$C(X,\mathbb R)$, such as character, tighntess, fan-tightness, strong
fan-tightness, the Fr{\'e}chet property and some of its variants. Under mild
conditions, local properties of $C(X,\mathbb R)$ at the zero function
correspond to the same property of the associated hyperspace structure at $X$.
The latter is often easy to characterize in terms of covering properties of
$X$. This way, many classical results are recovered or refined, and new results
are obtained. In particular, it is shown that tightness and character coincide
for the continuous convergence on $C(X,\mathbb R)$ and is equal to the
Lindel{\"o}f degree of $X$. As a consequence, if $X$ is consonant, the
tightness of $C(X,\mathbb R)$ for the compact-open topology is equal to the
Lindel{\"o}f degree of $X$.
