On a class of homeomorphisms of function spaces preserving the Lindelöf number of domains

We consider the class of all homeomorphisms between the function spaces of the form $C_p(X)$, $C_p(Y)$ such that the images of $Y$ and $X$ under their dual and, respectively, inverse dual mappings consist of finitely supported functionals. We prove that if a homeomorphism belongs to this class, then Lindelöf numbers $l(X)$ and $l(Y)$ are equal. This result generalizes the known theorem of A. Bouziad for linear homeomorphisms of function spaces.