I am trying to prove that $\left(a,b\right)$ is an open set in $\mathbb{R}$ under the usual metric $d\left(x,y\right)=|x-y|$

given any $x\in \left(a,b\right)$ I am supposed to produce an open ball which is contained in $\left(a,b\right)$.

I think $\epsilon=min\{x-a,b-x\}$ will work for us as the radius of the ball. That is $\left(x-\epsilon,x+\epsilon\right) \subset \left(a,b\right)$

I am unable to prove the following mathematically. $\left(x-\epsilon,x+\epsilon\right) \subset \left(a,b\right)$

Help me with this.

  • $\begingroup$ The desired containment is equivalent to $a \leq x - \epsilon$ and $x + \epsilon \leq b$, and these in turn follow immediately from your definition of $\epsilon$. $\endgroup$ – Bungo Sep 8 '18 at 5:50

enter image description here Maybe this small picture will help a bit.


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