Let $(X, \mathscr{T})$ and $\left(Y, \mathscr{T}^*\right)$ be topological spaces, let the function $f \colon (X, \mathscr{T}) \longrightarrow \left(Y, \mathscr{T}^*\right)$ be open and onto, and let $\mathscr{B}$ be a basis for $\mathscr{T}$. Then how to show that the collection $$ \big\{ f(B) \colon B \in \mathscr{B} \big\} \tag{0} $$ is a basis for $\mathscr{T}^*$?
My Attempt:
Let $V$ be any open set in $\left( Y, \mathscr{T}^* \right)$. We need to show that $V$ can be expressed as a union of some sets in the collection (0) above.
Since $f$ is onto, we have $$ V = f \left( f^{-1}(V) \right). $$
And, since $f$ is open, each set $f(B)$ in the collection (0) above is also an open set of $\left(Y, \mathscr{T}^* \right)$.
How to proceed from here?