On classic, Euclid, plane a pair of parallel lines would never intersect. Given a line and a point outside of the line, there can be only one line through the point which would be parallel to the first line.
If one modify the axiom to allow for many such lines through the point (the lines which would never intersect the first line) - that would result in hyperbolic plane geometry.
If one modify the axiom to state that no such line through the point is possible (i.e. any line through the point would intersect the original line) - that would result in the Riemann sphere geometry.