Metric Spaces and Topology

**CollegeMathKid** · October 13th 2009, 03:07 PM

A and B are any sets in a metric space (X, d)
Prove that
(A U B )' = A' U B'

' meaning interior.

And then prove that

(A U B ) closure = A closure U B closure

I know that you have to prove it both ways, and that the proofs will be relativley similar, and I know that they are true. I'm just really confused on where to get started

**Jose27** · October 13th 2009, 04:23 PM

Originally Posted by CollegeMathKid

A and B are any sets in a metric space (X, d)
Prove that
(A U B )' = A' U B'

' meaning interior.

This is not true: Take $X= \mathbb{R}$ , $A=(0,1) \cap \mathbb{Q}$ $B= (0,1) \cap (\mathbb{R} - \mathbb{Q} )$ then $int(A)=int(B)=\emptyset$ but $int(A \cup B)=(0,1)$