Given a map $f\colon (X,\tau_1)\rightarrow (Y,\tau_2)$ and an open $V\subset X,$ let $f_{\ast}(V)$ be the biggest open subset of $Y$ whose inverse image is contained in $V.$ Do you know any theorems of topology whose proof uses $f_{\ast}V?$
Motivation: I ask this question because Borceux says in his Handbook of Categorical Algebra Vol 3 page 17 that $f_{\ast}V$ does not play any significant role in topology. And, I couldn't recall any results which uses these sets.