[jerf]

The modern meaning of "geometry" may not change, but cutting and gluing space definitely break Euclidean geometry, as in, specifically the one defined by Euclid. You can't break a mathematical system much harder than to kill it at the axiomatic level. 4 out of 5, if not 5 out of 5, axioms do not hold if you include portals. That's pretty dead.

[zenorogue]

For an analogy: an Abelian group is a structure in which four axioms hold. A non-Abelian group is a structure in which three of these axioms hold, and the fourth does not. It is not a structure in which some random proper subset of axioms holds, because such a notion would be useless. While structures where specific subsets of axioms hold (non-Abelian groups, semigroups, monoids, etc.) are useful.