Definition

Consider a Topological Space $(X, T)$ and a Surjective map $p : X \to A$ . The quotient topology $T_{q}$ on $A$ consists of all sets $U \subset A$ whose pre-image $p^{- 1} (U)$ is Open Set in $(X, T)$ . $U \in T_{q} ⟺ p^{- 1} (U) \in T$