What "tools" exist within mathematics to visualize concepts for more advanced areas of mathematics, particularly within Analysis, Topology and Algebra ? Furthermore, can one rigorously integrate such "visual tools" within proofs?
As an example here are some that I've encountered so far:,
- I've seen commutative diagrams for compositions of maps.
- Another example is the following: If $X, W \subset \mathbf{R}^2$ are subsets of the plane, we can draw a picture of a map $f : X \to W$ by colouring $X$ with some pattern, and then colouring each point $(u,v)= f(x,y)$ by the same colour as $(x,y)$ as shown in the free book Stokes’s Theorem (Benjamin McKay) (on pages 3-4).
Edit: By visualization, I mean anything that conveys information diagrammatically or without the use of words, which hold true for the examples I provided.