Does the homoemorphism preserve the following: Let $F$ be a homomorphism between $U$ and $V$ where $U \subset X$ and  $V$ is open in the space $Y$.

Then is $U$ open in the space $X$?

If so how can we show this? 

thanks
Let $Y=\{a,b\}$ with the discrete topology and let $X=\mathbb{R}$. Letting $U=\{0\}$ and $V=\{a\}$, it is clear that $U$ is not open in $X$ even though $F:U\to V$ defined by $F(0)=a$ is a homeomorphism.
Take $X=\Bbb R, Y=[0,1),V=[0,\frac{1}{2})$.

Consider $F(x)=x$ then $F:U\to V$ is a homeomorphism but $U=[0,\frac{1}{2})$ is not open in $\Bbb R$