# Thread: Metric Spaces and Topology

1. ## Metric Spaces and Topology

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

2. 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 $\displaystyle X= \mathbb{R}$, $\displaystyle A=(0,1) \cap \mathbb{Q}$ $\displaystyle B= (0,1) \cap (\mathbb{R} - \mathbb{Q} )$ then $\displaystyle int(A)=int(B)=\emptyset$ but $\displaystyle int(A \cup B)=(0,1)$

3. 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.
That is false.
Let $\displaystyle A=[1,2]~\&~B=[2,3]$ then $\displaystyle 2\in(A\cup B)^o=(1,3)$.

BUT $\displaystyle 2\not\in A^o\cup B^o=(1,2)\cup(2,3)$.

4. However, you should be able to prove that $\displaystyle int(A)\cup int(B)\subseteq int(A\cup B)$.