# Open sets Proof

• Feb 1st 2010, 09:28 PM
jzellt
Open sets Proof
A) Show that the union of two nonempty open sets is open.

I have this theorem to use: A set D is open iff it contains no point of its boundary.

Can someone show have to give a solid proof of A...

Thanks in advance for any help
• Feb 2nd 2010, 11:08 AM
Here is what I think:

Say I have open sets X and Y, then for $\displaystyle \forall x \in X , \forall y \in Y$ , I can find an open ball centered at them such that the entire ball is in X and Y, respectively.

Well, then, if you pick any point in $\displaystyle X \cup Y$ , that that point is either in X or Y, well, then, you can draw another open ball that is contained in either X or Y.

So if you use radius $\displaystyle \epsilon , \delta$ for the points in X and Y, you can just use radius $\displaystyle min \{ \epsilon , \delta \}$

Hope this helps.
• Feb 2nd 2010, 12:11 PM
Drexel28
Quote:

Originally Posted by jzellt
A) Show that the union of two nonempty open sets is open.

I have this theorem to use: A set D is open iff it contains no point of its boundary.

Can someone show have to give a solid proof of A...

Thanks in advance for any help

Is this a topological space or a metric space? If it is the former than this is the definition. Otherwise, let $\displaystyle \left\{O_j\right\}_{j\in\mathcal{J}}$ be an arbitrary class of open sets in a metric space $\displaystyle X$. Let $\displaystyle x\in\bigcup_{j\in\mathcal{J}}O_j$ be arbitrary. Since $\displaystyle x\in O_k$ for some $\displaystyle k$ and $\displaystyle O_k$ is open there exists a $\displaystyle \delta>0$ such that $\displaystyle B_{\delta}(x)\subseteq O_k\subseteq\bigcup_{j\in\mathcal{J}}O_j$ and we are finished.