> It means it is no longer possible to draw a line between any two points (portals can shadow portions of the space such that some points are not connectible
This feels a bit like saying that putting a wall in a room makes it non-euclidean since there's now a barrier in the way.
I know where you're coming from, but this is confusing geometry and topology. (Curvature vs. how the space is connected.)
> This feels a bit like saying that putting a wall in a room makes it non-euclidean since there's now a barrier in the way.
Maybe that's what H. P. Lovecraft meant about non-Euclidean architecture in R'lyeh. ;D
(It probably isn't - the quote from The Call of Cthulhu says "the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres [...]".)
If you define "a line" as "blocked by another line (standing in for a wall)", it would be non-Euclidean, by virtue of breaking the axiom that says a line can be infinite extended, so that's not a particularly compelling argument.
I don't need to "confuse" geometry and topology. Euclid's axioms require both flat geometry and no "connections" in the space. It's right there in the axioms, if you can read them properly. They have no accommodations for "topology", and it is certainly not just a handwave "oh, whatever, it's no big deal" to extend them to handle it.
This feels a bit like saying that putting a wall in a room makes it non-euclidean since there's now a barrier in the way.
I know where you're coming from, but this is confusing geometry and topology. (Curvature vs. how the space is connected.)