This question may be a bit simple or even naive for some people but it indeed confuses me for a long time. Thank you all if you provide any explanation.
I know concepts: compactness means any open cover have finite subcover, which is equivalent to closed and bounded; connectedness means there's no disjoint decomposition by two nonempty open sets. However, I have no idea how they play roles in particular cases. I read many theorems that require compact and connected topology but there's no any mention in their proofs.
The situation occurs frequently, as far as I concern, in differential geometry and multivariable calculus (vector fields). Could anyone explain to me how they involve in mathematics? A few examples are better welcomed.
Please let me give some examples
Why do we consider compact Lie group in differential geometry? What role does compact here play here?
Why do we require compactness of manifold when we consider de Rham cohomology?