2
$\begingroup$

If I have a function defined as $f(x) = (\sin(2\pi x), \cos(2\pi x))$, where $f : (0,1) \to \mathbb{S}^1-\{(0,1)\}$, how should I go about showing it's surjective? Maybe I'm sleep deprived, I don't know, I feel like I keep going down a wrong path with inverse trig and it just seems... gross? Any tips?

Thanks!

  • 1
    Excuse me for asking, perhaps it is trivial: what is $\mathbb S^1$? – 2017-01-13
  • 0
    $\mathbb{S}^1$ is the unit circle, $\{\vec{x}\in\mathbb{R}^2 : \|\vec{x}\|=1\}$, – 2017-01-13
  • 0
    So we remove the point at the top of the circle in this problem – 2017-01-13

1 Answers 1

2

Given a point $(a,b)\in S^1\setminus\{(1,0)\}$, first look for $x$ such that $b=\cos(2\pi x)$. This gives you either $x=\frac{\arccos b}{2\pi}$ or $x=1-\frac{\arccos b}{2\pi}$.

Except for $(a,b)=(0,-1)$, these two values of $x$ will give different values $\sin(2\pi x)$ (namely, their signs will be different).

You also know that for a fixed $b$ there are at most two points on $S^1$ with this $b$ -- so those must be the values of $f$ from the two $x$s above. So no matter which of them is the $(a,b)$ you're looking at, it will be hit by $f$.

  • 0
    Do I need to mention anything about the preimage being a subset of the the open interval? Or am I fine with just showing the existence of a preimage for all points in the codomain? Just not sure how in-depth I need/have to be... – 2017-01-13
  • 1
    @BradFlynn: If what you want to prove is that $f$ is surjective, then that is all you have to show: each point in the codomain has at least one preimage. (The $x$ you find is of course not a preimage unless it is in the domain, but knowing the range of $\arccos$ will tell you that the $x$s you compute _are_ in the domain). – 2017-01-13
  • 0
    If we are going to prove continuity in a topological setting, how would you go about writing this? An open set in $\mathbb{S}^1 - \{(0,1)\}$ is a clockwise arc, based on our chosen function. So the preimage of this set is also an open set in $(0,1)$. Would it be necessary to show this "explicitly" or is this enough information for continuity? Like, just saying some point is "clockwise" greater than another is to say that their preimage point is greater than the other? – 2017-01-16