I was working on proving that $S^1$\{(1,0)} with the induced topology by $R^2$ ( product topology) is homeomorphic to the interval ]0,2π[..the homeomorphism is the function f(x)=(Cos(x),Sin(x)).. I already proved the bijection but I'm stuck proving the continuity .. Knowing that a function is continuous if the inverse image of every open set is open.
Can anyone help me.
Prove that this function is continuous
continuitygeneral-topology
Best Answer
To show that $f$ is a homeomorphism, we can use the following. If $f:X\rightarrow Y$ is a continuous bijection where $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism.
Then as @blamethelag suggested, you can prove continuity of $f$ by using results of product functions. (Here, I'll write $f(x)=(f_1(x),f_2(x))$.)
Let $U\times V$ be an open subset of $\mathbb{R}^2$. Then for every $x\in f^{-1}(U\times V)$, we have that $x\in f_1^{-1}(U)\cap f_2^{-1}(V)\subseteq f^{-1}(U\times V)$. Since $f_1$ and $f_2$ are continuous, $f_1^{-1}(U)\cap f_2^{-1}(V)$ is open in the domain of $f$, and this shows $f$ is continuous.