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This is part of the proof of Munkres Book.

Conversely, suppose $ $$ x = \left( {x_\alpha } \right) $$ $ lies in the closure of $ $ , in either topology ( box or product). We show that for any given index $ $$ \beta $$ $ we have $ $$ x_\beta \in \overline {A_\beta } $$ $ . Let $ $$ V_\beta
$$ $ be an arbitrary open set of $ $$ X_\beta
$$ $ containing $ $$ x_\beta
$$ $ . Since $ $$ \pi _\beta ^{ - 1} $$ $ applied to $ V$$ _\beta
$$ $ is open in $ $ in either topology, it contains a point y of $ $(this last part i don´t understand it , and sorry latex doesn´t work, but the proof is on page 116

I don´t understand why the property of being open in the product implies that exist that point, that´s my question )=

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