A neighbourhood in $\mathbb{R^2}$ can be an open ball of the form
$U(a, b) = ${${f(x, y): \sqrt{ (x - a)^2 + (y - b)^2} < r}$} for some $r>0 $.
However, it is also possible to have a rectangular neighbourhood of a point ($x_0,y_0$) such as:
($x_0-a,x_0+a$) x ($y_0-b,y_0+b$).
Does it make a difference to use one rather than the other? I found the second in the Implicit function theorem, but what does this imply? Why don't we use a general neighbourhood of undefined shape, since we don't know its boundaries?
Thanks in advance!