prove or disprove
Let $(X, \mathscr{T}_1)$ and $(Y, \mathscr{T}_2)$ be topological spaces and suppose that $f : X \to Y$ is a function that is $\mathscr{T}_1$ − $\mathscr{T}_2$ continuous. If $A \subseteq Y$ , then $\operatorname{int}(f^{-1}[A]) \subseteq f^{−1} [\operatorname{int}(A)]$
I think,it is False STATEMENT.
my work:
Take $X = Y = \mathbb{R}$ and let $f$ be the constant function sending everything to $3$. Let $A = \{3\}$. Then $\operatorname{int}(A) = \emptyset$ and so $f^{−1}[\operatorname{int}(A)] = \emptyset$. However, $f^{−1}[A] = \mathbb{R}$ and thus ,$\operatorname{int}(f^{−1}[A]) = \mathbb{R}$, but $\mathbb{R} \nsubseteq \emptyset$. I am not sure is it correct what I do.