Image of compact manifold over continuous function is compact?

Question

I'm sure it's true, because one should be able to cover the manifold with finite open covers and then use coordinate charts to reduce it to the Euclidean space case. I just want to make sure, I'm not overseeing something.

Kind regards.

Answer 1

There's no need to use coordinate charts. This is a basic theorem of topology, true for any continuous function from any compact topological space to any other topological space, and with a simple proof based on the definitions. I suggest you look it up in a topology book such as the one by Munkres.