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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.

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  • $\begingroup$ 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..... $\endgroup$
    – fleablood
    May 8, 2020 at 21:57
  • $\begingroup$ 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". $\endgroup$
    – fleablood
    May 8, 2020 at 22:01

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

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