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:

$\map p U$ is open in $X / \sim$

The saturation of $U$ is open in $X$

Proof

By definition of quotient topology, $\map p U$ is open in $X / \sim$ if and only if $\map {p^{-1} } {\map p U}$ is open in $X$.

$\blacksquare$

Also see

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