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let $X$ be a topological space. let $G$ be a group acting on $X$. let $x\in X$ with neighborhood $U_x$. let $\pi:X\rightarrow X/G$ the quotient map, we denote $[x]=\pi(x)$

1) is $\pi(U_x)$ a neighborhood of $[x]$? this means that the quotient map is always an open map.

2) if $A$ is a subset of $X$, is it true that $\pi(A)=A/G$?

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The answer to your first question is yes. If $G$ acts on $X$ continuously, the map $\pi:X\rightarrow X/G$ is open for $X/G$ endowed with the quotient topology. To see this, let $U$ be an open subset of $X.$ As $G$ acts continuously on $X$ and each element of $G$ is invertible, right multiplication by any element of $G$ is a homeomorphism of $X.$ Hence, $Ux$ is open for all $x\in G.$ It follows $\pi^{-1}(\pi(U)) = UG =\cup_{x\in G} Ux$ is open in $X.$ We conclude $\pi(U)$ is open in the orbit space $X/G.$

The latter of your questions isn't well posed. For example, what does $A/G$ mean if $A$ is not $G$ stable? However, in the case that $A$ is $G$ stable (i.e. $A = AG$) then the answer is yes $\pi(A) = A/G.$