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

**Runty** · September 15th 2010, 07:52 AM

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.

**Plato** · September 15th 2010, 10:43 AM

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 $A^o\cup B^o~\&~(A\cup B)^o?$

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

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