The definition of "circle" they are using is the set of points at an equal distance (the radius) from a given point (the centre). This definition works in any setting in which some sort of "distance" (metric) is defined. They are also using an implicit definition of "circumference" that works for the cases being considered here: split the "circle" into sections, measure the sum of their lengths according to the metric and take the limit as you use more and more, smaller and smaller sections. There are details that aren't covered in the article, but it works.

Having defined what a "circle" is and what its "circumference" and "radius" are, "pi" is defined: it's half the ratio of the circumference to the radius.

(I don't think it was very nice of whoever downvoted you, presumably because you're wrong, given you explicitly allowed that you might not be getting it.)

Careful, though. There isn't a constant "pi" for all metric spaces. Using distance along the surface of a sphere, the ratio of a circle's diameter to circumference depends on the size of the circle.

> (I don't think it was very nice of whoever downvoted you, presumably because you're wrong, given you explicitly allowed that you might not be getting it.)

I believe this is how this website works: if someone thinks you are wrong, they will downvote your comment. It's best not to think about in terms of niceness but more about getting the content most people agree with, or considered the most valuable by the majority, to the top so that more people can view and discuss it.