Is the Arbitrary Union of Open Sets in R Open?

[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.