A little confusion about compactness and connectedness
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 bounded close; 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.
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