# Decidability of intersection of two languages of same type

Given two context-sensitive languages, $L_1$ and $L_2$ is the problem of "whether $L_1 \cap L_2$ also belongs to CSL" decidable?

I have the same question for the case when $L_1$ and $L_2$ belongs to Recursive/Recursively enumerable language. I think, they are decidable because they are closed under intersection operation as can be seen here. But I haven't found any text mentioning this explicitly so I can't be sure.

• All three classes you mention are closed under intersection, so the problems are trivially decidable (i.e. by the algorithm: "return YES"). Are you perhaps searching for proof of closure under intersection? – potestasity Jan 18 '18 at 12:08
• I was looking for confirmation that closure under a property implies decidability for that property. I guess that is trivial then. @potestasity – momo Jan 18 '18 at 16:24