Does $A$ homeomorphic to $B$ imply $f^{-1}(A)$ is homeomorphic to $f^{-1}(B)$? Let $X$ and $Y$ be topological spaces, and $f:X\to Y$ a continuous map. Is the following true: 

If  $A$ and $B$ are two homeomorphic subspaces of $Y,$ then $f^{-1}(A)$ and $f^{-1}(B)$ are homeomorphic subspaces of $X$.


As an even simpler counterexample, take $f:\mathbb R\to\mathbb R$ with the usual topology:
$$f(x)= x^2$$
and take $A=[1,2]$ and $B=[0,1]$.
$A$ and $B$ are obviously homeomorphic, but $f^{-1}(A)$ consists of two separate intervals $[-\sqrt 2,-1]\cup[1,\sqrt 2]$ but $f^{-1}(B)$ is the single interval $[-1,1]$. Since one of the preimages is connected but the other isn't, they cannot be homeomorphic.
