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> I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$.
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For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ which is continuous and surjective. 

But how can I generalise this to n dimensions.