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
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.
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.