Oh hey maybe this would be a fun place to talk about my half-baked amateur #math and possibly #topology ideas!
Say I have a semi-metric space where for every point P and real distance D > 0, there is at least one point Q where distance(P,Q)=D.
Is that space interesting in any way? Does it have a name or set of named properties?
Earlier weblog entry (using a rule about accumulation points rather than the PQD thing): https://ceoln.wordpress.com/2024/02/09/semi-metric-woolgathering/
@ceoln I don't know if this qualifies as a fun response, but the first thing that comes to mind is that it can't be anything like a sphere on account of having to allow for arbitrary distances. (there's probably a projective geometry pro disproving me as they read this). whenever I try to imagine that only one point has to have distance \(d\) between another point, I end up imagining \(|\mathbb{R}|\) pairs of points with exactly that distance between them, so a bit of an uninteresting minmal example. I love feedback so fire away!
All responses are fun! :)
Can't be anything like a sphere in what sense? We can imagine all the points at distance r from C. Does that not turn out spherical enough?
Figuring out where our intuition goes entirely wrong without the triangle inequality is always interesting. 😁
Note that in the original there is at least one point at every D from *every* point.
But requiring that there is just at least one pair of points for every (real positive) distance sounds attractive, too.
I kind of think so? And/or dense or something 😆
@ceoln right? like if you tried to devise a space that wasn't connected that had all gaps in all the right places you'd end up making two copies of a dense space like the banach-tarsky paradox.
@ceoln Oh, I was thinking that in a sphere all geodesics sphere are finite. so if we're requiring that any \(d \in \mathbb{R}^+\) is allowed, then we have an infinitely large space.
There being at least one point at every D from *every* point sounds much cooler! It makes me wonder if the space must be continuous somehow.