Intersection of languages using closure properties

Question

1. $L_1 \in \mathrm{CFL} \cap L_2\in \mathrm{CSL}$ is $\mathrm{CSL}$ because every $\mathrm{CFL}$ is $\mathrm{CSL}$ and by applying closure property of $\mathrm{CSL}$ under intersection, it's $\mathrm{CSL}$.

2. $L_1 \in \mathrm{ RL} \cap L_2 \in \mathrm{CFL}$ is same as $\mathrm{CFL} \cap \mathrm{CFL}$ . As $\mathrm{CFL}$ is not closed under intersection, we promote $\mathrm{CFL}$ to $\mathrm{CSL}$. So $\mathrm{CSL} \cap\mathrm{CSL}$ is $\mathrm{CSL}$.

But, we know that 2nd is guaranteed to be $\mathrm{CFL}$ (also $\mathrm{CSL}$).

Why I am not able to arrive at the answer ($\mathrm{CFL}$) using closure properties for 2nd case ? Can't we apply closure properties when considering regular languages?

Answer 1

One of the closure properties of CFL is that they are closed onder intersection with regular languages. This can be directly applied, rather than first promoting the regular language to context-free language.

In general, if we want to know the family in which the intersection (or whatever operation $\oplus$) of two languages belongs, then we look for the smallest families $\mathrm{X}$ and $\mathrm{Y}$ that contain the two languages and for which we know an applicable closure property.

In your terminology, we "promote" as little as possible, but enough to find a family/families for which the closure property is known.