Let’s start with a Euclidean surface and examine what happens as we discard various properties. A two-dimensional Riemannian surface only includes intrinsic information, i.e. information that is independent of any outside structure, and so may not have a unique embedding in \({\mathbb{R}^{3}}\). For example, a sheet of paper is flat, and remains intrinsically so even if it is rolled up; i.e. there is no measurement possible on the surface of the sheet that can reveal whether it is rolled up or not. A plain manifold includes no notion of length at all, so the sheet of paper can be arbitrarily stretched, as long as there are no kinks or singularities introduced. There are other objects that we will not consider here, but may have application to physics. For example, an orbifold is a manifold where certain kinds of singularities are allowed.

Short of a manifold structure we can define a series of increasingly more primitive spaces, but they begin to include examples that are hard to reconcile with our intuitive idea of a “geometric object.” Our main purpose in studying these, in particular topological spaces, will be to determine what results only depend on “shape,” i.e. what results only depend on topology, as opposed to an additional manifold structure.