Generalizing surfaces

In algebra we added arithmetic properties to sets cumulatively until we had built enough structure to arrive at the real numbers. We then went further by considering generalizations of vectors. In geometry, we build up to a surface in three-dimensional space, and then go further by considering generalizations of tangent vectors to a surface.

In this section we will give a preview of the basic ideas, with the implicit assumption that the reader already has some familiarity with objects and relations that will only be defined later.

Added structure Resulting features
Topological space Open sets Connectedness, holes
Hausdorff space Disjoint neighborhoods Separation between points
Metric space Metric Distances between points
Topological manifold Cartesian charts Coordinates, dimension
Differentiable manifold Differentiable structure Tangent vectors, calculus
Riemannian manifold Riemannian metric Length, angles, volumes
Euclidean surface Embedding in \({\mathbb{R}^{3}}\) Relation to a higher space

Notes: There are many more messy details here than with algebra. These will be covered in the following chapters.

An Illustrated Handbook