Dual image map restricts to open sets?

You’re right that there’s a problem here.

Let $A=(0,1)times(0,1)$ and $B=(0,1)$, each with the usual topology, and let $$f:Ato B:langle x,yranglemapsto x$$ be the projection to the $x$-axis; this is a continuous, open map. Let $$U={langle x,yranglein A:y>2x-1};.$$ $U$ is open in $A$, but

$$f_*(U)=left(0,frac12right];,$$

which is not open in $B$.

You do get the result if $f$ is closed and continuous. In that case let $U$ be open in $A$, and let $F=Asetminus U$. Suppose that $bin B$; then $f^{-1}[{b}]subseteq U$ iff $bnotin f[F]$, i.e., iff $bin Bsetminus f[F]$, so $f_*(U)=Bsetminus f[F]$, which is open in $B$.