How can there exist a quotient map to a space of higher dimension than the domain?

Solution 1:

It’s just an error, by you or by your professor. The projective plane results from identifying antipodal points on $S^2.$

However, there do exist counterintuitive quotient maps that increase dimension. A space-filling curve is a continuous surjection $[0.1]\to [0,1]^2$; since the domain is compact and the codomain is Hausdorff, this is also a closed map, thus in particular, a quotient map. (This cannot happen with differentiable functions, or with injective continuous functions, happily.)

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