Proving that $∂U = \overline U \cap \overline{X\setminus U}$

Question

I want to prove that $$∂U = \overline U \cap \overline{X\setminus U}.$$

Not too sure where to start with this question so any help would be appreciated.

I am also struggling to prove $U^{\circ} \subset U$ for any $U$. I have the following definitions: $$∂U = \overline U \setminus U^{\circ},\ U^{\circ} = X\setminus \overline{(X\setminus U)}.$$

Any help is much appreciated.

Answer 1

With the given definition of interior, we have:

$$∂U = \overline U \setminus U^{\circ} =\overline U \setminus( X\setminus \overline{(X\setminus U)}) = \overline U \cap \overline{(X\setminus U)}.$$

Also:

$$ U^{\circ} =X\setminus \overline{(X\setminus U)} \subset X\setminus (X\setminus U) = U, $$ as $X\setminus U \subset \overline{(X\setminus U)}$.