Which branch of topology to study

[shounakbhatta]

Hello,

I learned that there are 4 types of approach to topology:

(1) General
(2) Algebraic
(3) Differential
(4) Geometrical

To have a rough understanding of General relativity, which branch of topology should I study?

Thanks.

 

[martinbn]

To have a rough understanding you need very little topology. What you need to study is differential geometry.

 

[dextercioby]

Oh, dear. You're thinking way too far. General Relativity is mathematically founded on differential geometry and advanced calculus. To study differential geometry at a decent level, you don't need too much topology (certainly not spitted per branches).

 

[pervect]

The foundational elements of topology are of some interest, Wolfram mathematics calls this "point set topology". This is the part of topology before you split off into branches.

Point-Set Topology

The low-level language of topology, which is not really considered a separate "branch" of topology. Point-set topology, also called set-theoretic topology or general topology, is the study of the general abstract nature of continuity or "closeness" on spaces. Basic point-set topological notions are ones like continuity, dimension, compactness, and connectedness.

Wolfram doesn't mention the definition of a manifold as part of point-set topology though that's something you might want to know :-). (I'm think it would be classified as part of point-set topology, but I could be mistaken as my focus is on the physics. Wolfram doesn't mention it.)

Wald, "General Relativity", for instance, has enough point set topology to give you a definition of a manifold in one of his appendices. A full study of it would probably be overkill if you are into the physics rather than the math.

 

[sardar]

I would recommend studying differential topology for a better understanding of general relativity. This branch of topology deals with smooth, continuous functions and their properties, which are essential in understanding the curvature of spacetime in general relativity. It also involves the study of manifolds, which are used to model the geometry of spacetime in general relativity. Additionally, studying differential topology can also provide a deeper understanding of the mathematical concepts and tools used in general relativity, such as tensors and differential equations. However, it is important to note that a thorough understanding of all branches of topology can contribute to a comprehensive understanding of general relativity.