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.