Two closed or open subsets in a Tychonoff space

Question

Let $ C_1 $ and $C_2$ be two closed subsets in a Tychonoff space $X $ (it is hausdorff). I am looking for an equivalent condition for $int (C_1)\subseteq C_2$, where $int (A) $ means the interior of the subset $A$ in $X $. And also

Let $ U_1 $ and $U_2$ be two open subsets in a Tychonoff space $X $ (it is hausdorff). I am looking for an equivalent condition for $cl (U_1)\subseteq U_2$, where $cl (A) $ means the closure of the subset $A$ in $X $.

Answer 1

You may know this. For improvement you should add why do you need it, or in which context you want to use it.

$int(C_1)\subset C_2\iff int(C_1)\subset int(C_2)\iff(\forall U\in\tau,U\subset C_1\implies U\subset C_2)\iff {C_2}^c\subset int(C_1)^c\iff int(C_1)\cap int(C_2)^c=\emptyset$