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.)
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 [...]".)
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.