Indestructibly productively Lindel\"of and Menger function spaces

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 $C_{\lambda }(X)$.