# How to define close points in plane?

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.

• Well, "close" is subjective. My buddy Andy Ant doesn't think $(0,0), (1,0)$ and $(-1,1)$ are very close together at all. $(0,0)$ and $(1,0)$ are an entire unit apart and $(1,0)$ and $(-1,1)$ at an inconceivably distance of $\sqrt 3$ units apart! That's almost TWO effing units!!!! Meanwhile my pal Willy Whale says $(0,0),(1,,0),(-1,1)$ and $(100,100)$ are extremely close together. $(100,100)$ is the only one that is even noticebly separate from the other but as its less then 200 measly units it hardly counts. So.... to be continued..... May 8, 2020 at 21:57
• To be close can only mean to be within a specified small distance from each other. And that is formally defined as .... being within a specified small distance from each other. i.e The points of $A = \{(0,0),(1,0), etc\}$ are closed together if $\sup \{d(x,y)| x,y \in A\} < \delta$ for some arbitrary $\delta$ that you declare to be "small". May 8, 2020 at 22:01

"Close" generally means "within distance $$\delta$$ of each other", where $$\delta$$ is some small positive number.