Definition:Quotient Topology

Definition
Let $$\left({X, \vartheta}\right)$$ be a topological space.

Let $$\mathcal R \subseteq X^2$$ be an equivalence relation on $$X$$.

Let $$q_\mathcal R: X \to X / \mathcal R$$ be the quotient mapping induced by $$\mathcal R$$.

Let $$\vartheta_2$$ be the identification topology on $$X / \mathcal R$$ by $$q_\mathcal R$$:
 * $$\vartheta_2 = \left\{{U \subseteq X / \mathcal R: q_\mathcal R^{-1} \left({U}\right) \in \vartheta_1}\right\}$$.

Then $$\vartheta_2$$ is called the quotient topology on $$X / \mathcal R$$ by $$q_\mathcal R$$.

Quotient Space
Thus we have that $$\left({X / \mathcal R, \vartheta_2}\right)$$ is a topological space.

It is called the quotient space of $$\left({X, \vartheta_1}\right)$$ by $$\mathcal R$$.

See quotient space.