Although this question might sound a little too simple, it is a problem that I must get addressed. In addition, there is no way for me to formally describe it. If you have something you can add, by all means; add it to my question to help me gain the answer I am looking for.

After pondering over this following definition on $\epsilon$-balls, I was not able to grasp the intuition for $\epsilon$ generally speaking:

Definition ($\epsilon$-balls). Let $(X,d)$ be a metric space. For each point $x\in{X}$ and each real number $\epsilon>0$, let $$B_d(x,\epsilon)=\{y\in{X}:d(x,y)<\epsilon\}$$ be the $\epsilon$-ball around $x$ in $(X,d)$.

For this definition and any other topological definition involving the universal notion of $\epsilon$, I was not able to grasp its effect and meaning and how it affects the definition. I am also trying to come up with an example pertaining to the above definition but cannot think of any.

Moving forward, I want an intuition for the use of $\epsilon$ for topology and other higher mathematics. I see $\epsilon$ is numerous important proofs, definitions and theorems. And if I do not know how to use them or understand why/how they are there, I will be in trouble when it comes to more complicated subject matters.

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    $\begingroup$ There is nothing to this $\varepsilon$, you could just call it $b$ or $\beta$ or $\mathfrak{JULIAN}$ or whatever. it just stands for the radius. so it is often called $r$ or $R$ or $\mathcal{R}$. You should read some $\varepsilon-\delta$ proofs, as they are called, then you will understand how you use it $\endgroup$ – Mister Benjamin Dover Feb 2 '15 at 23:25
  • $\begingroup$ Not quite sure if that's what you have in mind, but $\epsilon$ (and $\delta$) are usually imagined to be very very small quantites. That's just a convention, though - you could replace $\epsilon$ by any variable name really. $\endgroup$ – AlexR Feb 2 '15 at 23:26
  • $\begingroup$ Not sure if it's what you're asking, but $\epsilon$ is a number. That's it, nothing else. The $\epsilon$-ball centred at $x$ is the set of all points whose distance from $x$ is less than $\epsilon$. For example in $\Bbb R^2$ with the usual metric, a $1$-ball is just an open disc of radius $1$. $\endgroup$ – David Feb 2 '15 at 23:27
  • $\begingroup$ As you can see from the three more-or-less differing answers in these comments, you should consider clarifying your question. $\endgroup$ – AlexR Feb 2 '15 at 23:28
  • $\begingroup$ @Laters However, it says "for all $\epsilon$" and and my thinking is that it does not work "for all". Could you provide me with an example as described in my question? $\endgroup$ – Julian Rachman Feb 2 '15 at 23:28

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