We have $f^{-1} \emptyset = \emptyset$ which is open in $T_X$ by definition. Next, by definition, a map is a
D359: Left-total binary relation and therefore
\begin{equation}
f^{-1} Y
= \{ x \in X : f(x) \in Y \}
= X
\end{equation}
which is again open in $X$ by definition. The claim now follows from result
R324: Continuity characterised by preimages of open sets. $\square$
