If one draws a figure on a flat inflexible plane than crumples it up, the length of that line remains invariant as does the order of the points on the line. If one allows the surface to also stretch the length changes yet the order of points remain the same. Seems to me allowing stretching is somehow cheating. Can someone clear my muddled beginner student thoughts on this? Thanks.

Mathematics studies both kinds of transformations. Isometries preserve distances, while homeomorphisms allow stretching, but forbid glueing and tearing. Each kind of transformations has its properties and theorems. Neither kind is cheating.