Does a neighbourhood have to extend from the point $a$ to both directions equally?

Question

The $\epsilon$-neighbourhood of a point "$a$" in $\mathbb R$ is defined as the set of points $x$ about a point "$a$" such that $|x-a|<\epsilon$, $\epsilon>0$.

But this means that the point "$a$" will be in the middle of the neighbourhood?

Answer 1

No, not at all. A neighborhood $N$ of $a$ must contain some interval $(a-r,a+r)$, with $r>0$. But $N$ doesn't have to have a middle point and, even if it has one, the middle point doesn't have to be $a$. For instance, $(1,+\infty)$ is a neighborhood of $2$.

Answer 2

Your instinct to investigate this idea is a good one. As the subject develops, there are distinctions to be made, and the idea of a neighbourhood of a point becomes any open set containing it. A collection of open sets with appropriate properties is called a topology, and can be defined without any reference to distance.

What you have defined is best called an open ball around $x$ - because it is an open set containing $x$ it is a neighbourhood of $x$ and remains a neighbourhood as the definition and context change - it is the classic example of a neighbourhood in some ways. The definition of a ball does put $x$ at the centre.