# How can I prove that there exists a surjective continuous map?

I was trying to prove that there exists a continuous function from a space to many others, and to see in what cases there exists such a function.

Clearly, the first things to note are properties preserved by continuous functions, like compactness and connectivity. Are there more general criteria?

I tried to prove or disprove the existence of a continuous function from the sphere (the boundary) to the circle (also the boundary).

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It's easy to see that there is a surjective continuous map from $[0,1]$ to $\mathbb S^1$. Can you see how to make a continuous surjection from $\mathbb S^2$ to $[0,1]$? – JSchlather Aug 30 '11 at 18:00
the construction of such function ( S^2 to [0,1] ) it´s geometrically? or it´s only a non special function? Dx? – Daniel Aug 30 '11 at 18:54
@Daniel: e.g. $(x,y,z) \mapsto |z|$. Quite geometrical I'd say :) – t.b. Aug 30 '11 at 19:03
If you want the most general answer, it will have to be "A surjective continuous mapping exists from $X$ onto $Y$ if and only if there is a function from $X$ which is onto $Y$ and the preimage of every open set in $Y$ is open in $X$". Of course if you have some limitations to pose the answer can be less tautological and with more content in it. – Asaf Karagila Aug 30 '11 at 22:20