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If we consider the open set $\mathbb{R}$, then for every $a \in \mathbb{R}$, you can find an open interval $(a-\epsilon, a+\epsilon)$.

I am probably over thinking this, but I am wondering: Why would it be an open interval if the boundary points are elements in $\mathbb{R}$? I know that $\mathbb{R}$ is both closed and open, but I don't see how the intervals would be open.

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    Okay, that helps clarify things, thank you both!2012-12-07

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An open interval $(a,b)$ is open because for every $x\in (a,b)$, there is an $\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\subset (a,b)$.

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    Yes, I was just over analyzing the idea of the open ball being "open" when it could be a closed interval. I got it now, thanks.2012-12-07