$f^{-1}(U)$ is regular open set in $X$ for regular open set $U$ in $Y$, whenever $f$ is continuous. Let $f$ be a continuous function from space $X$ to space $Y$. If $U$ is regular open set in $Y$, it it true that $f^{-1}(U)$ is a regular open set in $X$?

Not necessarily.  Consider the absolute value function $x \mapsto | x |$, and the inverse image of $(0,1)$.
