Is there a continuous and injective function $f: [0,1] \to \mathbb R^2$ such that the image has positive two dimensional Lebesgue measure?
I am not sure how to approach here. I think ine way to approach this would be by contradiction. I think that we must use the fact that a contiuous bijective map from a campact set to a Hausdorff set is a homeomorphism. However, I am not sure what to do here. If only the property of being positive Lebesgue measure were a topological property...