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.