Show that a homeomorphism is an open map

Question

I am struggling to show that a homeomorphism maps an open set to an open set. I have seen online the identity $$f(U) = (f^{-1})^{-1}(U)$$ but I get confused since consider $f(x)=2x$. $f(1,2)=(2,4)$ but $(f^{-1})^{-1}(1,2) = (1,2)$. Can someone please point out where I am getting confused.

Answer 1

Since, $f(x)=2x$, $f^{-1}(x)=\dfrac x2$. Therefore,\begin{align}(f^{-1})^{-1}\bigl((1,2)\bigr)&=\left\{x\in\mathbb R\,\middle|\,f^{-1}(x)\in(1,2)\right\}\\&=\left\{x\in\mathbb R\,\middle|\,\frac x2\in(1,2)\right\}\\&=(1,4)\\&=f\bigl((1,2)\bigr).\end{align}