Basics of Topology and Differential Geometry

References:

•  Interactive Real Analysis  : a very complete online textbook from sets to differentiation with java tools (root finder and an awesome function plotter)
• Topology by Klaus Janich, Springer-Verlag ed. : undergrad textbook to learn the fundamentals in topology
• Topology, a first course by James R. Munkres, Prentice Hall ed. : far than enough for a computer graphist.

Topology "with the hands"

1. Open set, closed set in Rⁿ

An open set "does not reach its bound" but comes infinitely close to. A closed set does.
 

 

2. Connected, convex properties straightforward in Rⁿ. 

1. Rⁿ is complete with usual metrics (Cauchy sequences converge).

A sequence converge if d(x_i , limit) -> 0 on your surface. A Cauchy sequence if such that d(x_i , x_j ) -> 0 when i and j goes to infinity but they may not converge to a limit. In a complete space only they do and any Cauchy sequence converge. Counter-example: an open set in Rⁿ , with a sequence that goes infinitely close to the "boundary". We will consider the space as coomplete after this point.
 
For fanatics, completeness give very useful results with series, see link.
In a complete space, Hausdorff distance and fixed point theorem are the basis of Fractal theory, which lead for instance to fractal image compression or to Julia sets.

 

2. Continuity

You don't need to be Super Mario to navigate on your surface.
If a sequence converges, then it converges also on your surface: (x_i ) cv => f(x_i ) cv

  ----> 
 
 

 

3. Differentiability

We can define a tangent at every point: a line in 2D, a plane in 3D, a hyperplane in higher dimensions.

For Implicit functions, the normal vector is defined by the gradient:  

For Explicit functions, the normal vector is, for a single linear map:

 
 
 
4. Related operators: Div, grad, curl and all that... thanks to read the book by Schey.

 
5. C0, C1, C2, Cinf: higher degrees of differentiability

 
Need to consider differentials as linear functions and iterate the processus. Tied to differentiability of partial derivatives, even if there is some subtilities you have to watch. See reference books.
 
 
 

Huge difference between continuous and discrete.