# Math Help - Prove that A^0 U B^0 is a subset of the interior of A U B

1. ## Prove that A^0 U B^0 is a subset of the interior of A U B

Let $A$ and $B$ be subsets of $\Re^n$ with $A^0$, $B^0$ denoting the sets of interior points for $A$ and $B$ respectively. Prove that $A^0\cup B^0$ is a subset of the interior of $A\cup B$. Give an example where the inclusion is strict.

2. Originally Posted by Runty
Let $A$ and $B$ be subsets of
$\Re^n$ with $A^0$, $B^0$ denoting the sets of interior points for $A$ and $B$ respectively. Prove that $A^0\cup B^0$ is a subset of the interior of $A\cup B$. Give an example where the inclusion is strict.
Recall that the interior of a set is the largest open subset of a set.
The union of two open sets is open. So what about $A^o\cup B^o?$
Is that union a open subset of $A\cup B?$

In $\mathbb{R}^1$ let $A=[0,1]~\&~B=[1,2]$ what are $A^o\cup B^o~\&~(A\cup B)^o?$