Quotient Topology is Topology

Theorem

Let $\struct {S, \tau}$ be a topological space.

Let $\RR \subseteq S \times S$ be an equivalence relation on $S$.

Let $q_\RR: S \to S / \RR$ be the quotient mapping induced by $\RR$.

Let $\tau_\RR$ be the quotient topology on $S / \RR$ by $q_\RR$.

Then $\tau_\RR$ is a topology on $S$.

Proof

By definition of quotient topology, $\tau_\RR$ is the identification topology on $S / \RR$ with respect to $q_\RR$ and $\struct {S, \tau}$.

Identification Topology is Topology shows that $\tau_\RR$ is a topology on $S / \RR$.

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.): $2$: Topological Spaces and Continuous Functions: $\S 22$: The Quotient Topology