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

2. Originally Posted by Runty
Let $\displaystyle A$ and $\displaystyle B$ be subsets of
$\displaystyle \Re^n$ with $\displaystyle A^0$, $\displaystyle B^0$ denoting the sets of interior points for $\displaystyle A$ and $\displaystyle B$ respectively. Prove that $\displaystyle A^0\cup B^0$ is a subset of the interior of $\displaystyle A\cup B$. 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 $\displaystyle A^o\cup B^o?$
Is that union a open subset of $\displaystyle A\cup B?$

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