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Quotient map from $X$ to $Y$ is continuous and surjective with a property : $f^{-1}(U)$ is open in $X$ iff $U$ is open in $Y$.

But when it is open map? what condition need?

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A quotient map $f \colon X \to Y$ is open if and only if for every open subset $U \subseteq X$ the set $f^{-1} (f (U))$ is open in $X$. A sufficient condition is that $f$ is the projection under a group action. –  Andrea Sep 1 '11 at 13:40
    
anything that is locally a projection should be open –  yoyo Sep 1 '11 at 22:21
    
Another sufficient condition: that the map be a surjective submersion of smooth manifolds. –  Bruno Stonek May 17 '13 at 16:48
    
@Andrea: "A sufficient condition is that f is the projection under a group action" Why, please? –  Ian Mallett Oct 26 '13 at 21:45
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If $\pi \colon X \to X/G$ is the projection under the action of $G$ and $U \subseteq X$, then $\pi^{-1} (\pi (U)) = \cup_{g \in G} g(U)$. –  Andrea Oct 27 '13 at 14:54

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