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 $\mathcal R$.

Also see

• Quotient space

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Functions