Projection of Subset is Open iff Saturation is Open
Theorem
Let $\sim$ be an equivalence relation on a topological space $\struct {X, \tau}$.
Let $p$ denote the quotient mapping induced by $\sim$.
Let $\tau_\sim$ be the quotient topology on $X / \sim$ by $p$.
Let $\struct {X / \sim, \tau_\sim}$ be the quotient space of $X$ by $\sim$.
Let $U \subset X$.
Then the following are equivalent:
- $\map p U$ is open in $\struct {X / \sim, \tau_\sim}$
- The saturation of $U$ is open in $\struct {X, \tau}$
Proof 1
By definition of quotient topology, $\map p U$ is open in $\struct {X / \sim, \tau_\sim}$ if and only if $\map {p^{-1} } {\map p U}$ is open in $\struct {X, \tau}$.
$\blacksquare$
Proof 2
Neccessary Condition
We are given that:
- $\map p U$ is open in $\struct {X / \sim, \tau_\sim}$
So:
- $\map p U \in \tau_\sim$
We are given that $p$ is the quotient mapping induced by $\sim$, so:
- $\overline U = p^{-1} \sqbrk {p \sqbrk U} \in \tau$
Hence:
- The saturation of $U$ is open in $\struct {X, \tau}$
$\Box$
Sufficient Condition
We are given that:
- The saturation of $U$ is open in $\struct {X, \tau}$
We are given that $p$ is the quotient mapping induced by $\sim$, so:
- $\overline U = p^{-1} \sqbrk {p \sqbrk U} \in \tau$
So:
- $\map p U \in \tau_\sim$
Hence:
- $\map p U$ is open in $\struct {X / \sim, \tau_\sim}$
$\Box$
Hence the result
$\blacksquare$