TITLE: prove that the closure of the intersection of A and B is the subset of the intersection of the closure of A and the closure of B.
QUESTION [6 upvotes]: prove that the closure of the intersection of A and B is the subset of the intersection of the closure of A and the closure of B.
my proof: let x be in cl(A intersects B), then x is in the intersection of A and B and x is in the set of the limit of the intersection of A and B.
case i: x is in cl(A intersects B), then x is in A and x is in B, then x is in the closure of A and x is in the closure of B. Therefore, x is in the intersection of the closure of A and the closure of B.
For the second part, I'm not sure how to do it. Please help! Thank you.
I'm using the Topology without tears book. (3.2 #2) It's online.

REPLY [2 votes]: We know that A $\cap$ B is a subset of A, and A $\cap$ B is a subset of B. Also, A is a subset of cl(A) and B is a subset of cl(B) (since cl(X) is the smallest closed set containing the set X).  Thus, we can conclude that A $\cap$ B is a subset of each of cl(A) and cl(B).  Therefore, cl(A $\cap$ B) is a subset of cl(A) $\cap$ cl(B).