Prove that $(A \cap B)^\circ = A^\circ \cap B^\circ$ Let $(X,d)$ be a metric space and let $A, B \subset X$  
How can I show that  $(A \cap B)^\circ = A^\circ \cap B^\circ$ ?
Please just tell me a Hint. ($A^\circ$ and $B^\circ$ are sets of interior points of A and B)

One definition of $A^\circ$ is the largest open set contained in $A$.
Using this definition, you can note that $A^\circ \cap B^\circ$ is an open set contained in $A \cap B$.
The other direction is similar but note that $A^\circ \cap B^\circ$
is an open set contained in both $A$ and $B$.
