Projection of Subset is Open iff Saturation is Open

Theorem
Let $\sim$ be an equivalence relation on a topological space $X$.

Let $X/\sim$ be the quotient space.

Let $p$ denote the quotient mapping.

Let $U\subset X$.

Then the following are equivalent:
 * $p(U)$ is open in $X/\sim$
 * The saturation of $U$ is open in $X$

Proof
By definition of quotient topology, $p(U)$ is open in $X/\sim$ $p^{-1}(p(U))$ is open in $X$.

Also see

 * Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open