Curves are continuous maps from an interval to a topological space. So a natural question is: is there any surface whose image is not the image of a continuous surjection from a rectangle to a topological space? Wikipedia's page about surfaces says "(...) there are surfaces for which there cannot exist a single parametrization that covers the whole surface." but doesn't give any example...
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the 2-sphere is an example – janmarqz Mar 15 '21 at 17:56
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@janmarqz how to prove that? that would answer this question: https://math.stackexchange.com/q/4062600/881385 – Carla is my name Mar 15 '21 at 18:12