  For a Tychonoff space $X$ and a family $\lambda$ of subsets of $X$, we denote
by $C_{\lambda}(X)$ the $T_1$-space of all real-valued continuous functions on
$X$ with the $\lambda$ -open topology. A topological space is productively
Lindel\"of if its product with every Lindel\"of space is Lindel\"of. A space is
indestructibly productively Lindel\"of if it is productively Lindel\"of in any
extension by countably closed forcing. In this paper, we study indestructibly
productively Lindel\"of and Menger function space $C_{\lambda}(X)$.
