I was studying on Follan's Real Analysis book and in Chapter 5.3, I saw the Open Mapping Theorem which stated (with some adjustment) as follows:
Let $X,Y$ be Banach spaces and $T:X\to Y$ be linear and bounded operator. If $T$ is surjective, then $T$ maps open sets to open set.
Then, based on the expression above, I expect if $T$ is nonlinear then the conclusion must fail. So I consider an example as follows: Let $X=Y=\mathbb{R}$ be (simplest?) Banach spaces and then take a mapping $f: \mathbb{R} \to \mathbb{R}$ with $f(x) = x^2$. Then consider open interval $U:=(-1,1)$ then $f(U) = [0,1)$ which is not open. Is my thinking correct?