Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?


share|improve this question
add comment

1 Answer 1

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 $\mathbb R/\sim$.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.