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.)
zavier
·
2021-12-20T10:04:42Z (almost 3 years ago)