Union of open sets question

[autre]

I have to prove that the arbitrary union of open sets (in R) is open.

So this is what I have so far:

Let $\{A_{i\in I}\}$ be a collection of open sets in $\mathbb{R}$. I want to show that $\bigcup_{i\in I}A_{i}$ is also open...

Any ideas from here?

 

[Deveno]

what is your definition of open set?

 

[autre]

The definition we use is that a set $A\subseteq\mathbb{R}$ is an open set if for each $x\in A$ there exists an $\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\subseteq A$.

 

[Deveno]

note that if $x \in \bigcup_{i \in I}A_i$, then necessarily $x \in A_i$ for some i.

can you continue...?

 

[autre]

Let $\{A_{i\in I}\}$ be a collection of open sets in $\mathbb{R}$. Let $x\in\bigcup_{i\in I}A_{i}$, then $x\in A_{i}$ for some $i$. Since each $A_{i}$ is open, there exists an $\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\subseteq A_{i}\subseteq\bigcup_{i\in I}A_{i}$. Thus, $\bigcup_{i\in I}A_{i}$ is open...

Am I on the right track?

 

[Deveno]

looks good to me.