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I'm taking some maps from quotient spaces to prove the continuity. I'm thinking if I can use the characteristic property to prove the continuity of the map. For example, take the map $f:\mathbb R/\sim \to \mathbb S^1$, where the equivalence relation is given by $x\sim y$ iff $x$ and $y$ differ by an integer, so I take this map: $f:\mathbb R/\sim \to \mathbb S^1$ defined by $f([x])=(\cos(2\pi x),\sin(2\pi x))$. So the question is if I prove that this map $g:\mathbb R \to \mathbb S^1$, $g(x)=(\cos(2\pi x),\sin(2\pi x))$ is continuous, then $f$ is continuous? Please anyone can help?

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up vote 6 down vote accepted

This is so by the universal property of quotient space. $A/B\to C$ is continuous iff its composition $A\to A/B\to C$ with the canonical map is.

Or: The inverse image under $x\mapsto(\cos(2\pi x),\sin(2\pi x))$ of an open set in $S^1$ is an open set in $\mathbb R$, which is $\sim$-invariant by the properties of $\sin$ and $\cos$, hence "is" an open set of $\mathbb R/\sim$.

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