Prerequisites to study cohomology?

Question

Work related I have to deal with cohomology theory fairly soon. Unfortunately, I never had any classes on this, so I'd like to study it on my own. Before I dive into a book or two, I'd like to make sure that I have all required previous knowledge to actually understand it. It would be great if someone could give a list of topics one should chronologically cover in order to be prepared to attack cohomology. (In order to make sure that no topic is omitted, imagine this question is asked by a high-school student, who never had any advanced math.) Thanks for any suggestion!

Answer 1

A. De Rham cohomology.

i) some general topology (basic notions: what topological spaces are, compactness, connectedness) ii) some smooth manifold theory (basic notions: what manifolds are, tangent spaces) iii) some linear algebra (basic notions: vector spaces, exact sequences, quotient spaces)

B. Sheaf cohomology.

i) some sheaf theory (basic notions: what they are, left exactness of the global sections functor) ii) some homological algebra (derived functors)

C. Singular cohomology i) some general topology (basic notions: what topological spaces are, compactness, connectedness) ii) some linear algebra (basic notions: vector spaces, exact sequences, quotient spaces)

These prerequisites are minimal in the sense that they allow you to understand the definitions, but you will of course need more to understand interesting/advanced results.