What I am looking for is a formalization of the intuitive ideia of closeness of points in a plane. I don't know if this formalization already exists.
For example, how can I rigorously define that $(0,0)$, $(1,0)$ and $(-1,1)$ are points close to each other in the plane, but $(0,0)$, $(1,0)$, $(-1,1)$ and $(100,100)$ are not? My question is in general, not for this particular example.
I tried thinking about minimum polygonal area that contains all points, but without success.
What I wanted was some ideia similar to limit in Analysis. Like, there is an intuition about convergence of a sequence, but it is important to be rigorous using the epsilon definition.