How do mathematicians measure area for points?
There are two points (adjacent or not adjacent) on top of an infinite plane; how to measure how much area they catch together, as a "set".
Because that if we put a minimal border around the two points that border would be made from ten points by itself and would catch its own ten-point-area, it won't be clear what is really a part of what:
- One set for all points together
- Two different sets: One for the two points and one for the border points
1 answer
The area is a kind of measure. In mathematics a measure is a function satisfying three basic axioms (properties). You can have a look at https://en.wikipedia.org/wiki/Measure_(mathematics)#Definition for an introduction and an explanation of the needed axioms.
Usually (if a different meaning isn't specified) "plane" means R^2, and "area" means the Lebesgue two-dimensional measure, which values 0 on any finite set (two points are an example of a finite set).
However there is no reason why you shouldn't define a different measure on your set, if that suits your (mathematical) purposes; if you use e.g. the "counting measure" (https://en.wikipedia.org/wiki/Measure_(mathematics)#Instances) your "area" of two points would be 2.
(NB: your initial assertion that the points may be "adjacent" may hint at the fact that you may not be using the standard topology, in particular one which is not T2. This is most probably not what you mean.)
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