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so finding a surjective map f is equal to finding a surjective map f':D^n →D^n/S^n and that is quotient map.

Now if I take now a continuous surjective function [0,1]→[0,1]^n e.g Peano curve and I compose the continuous surjective function [0,1]^n→S^n above I should have the proof .

the second one I founnd on internet is this A non-empty Haussdorff topological space is a continuous image of the unit interval if and only if it is a compact, connected,locally connected andsecond countabke space , S^n has all this properties so there exist a continuous and surjective function from [0,1]^n→S^n

Are the 2 proofs correct?