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.