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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.

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If I'm not mistaken such sets are used to define the direct image sheaf. But perhaps that doesn't count as topology... –  Zhen Lin Jul 1 '11 at 15:31
    
The direct image sheaf is simpler than that, no? $(f_*\mathscr{F})(U) = \mathscr{F}(f^{-1}(U))$. –  Dylan Moreland Jul 1 '11 at 16:19
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