Definition:Quotient Topology

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Let $T = \left({S, \tau}\right)$ be a topological space.

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

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

Definition 1

Let $\tau_\RR$ be the identification topology on $S / \RR$ by $q_\RR$:

$\tau_\RR := \set {U \subseteq S / \RR: q_\RR^{-1} \sqbrk U \in \tau}$

Then $\tau_\RR$ is the quotient topology on $S / \RR$ by $q_\RR$.

Definition 2

Definition:Quotient Topology/Definition 2

Quotient Space

The quotient space of $S$ by $\RR$ is the topological space whose points are elements of the quotient set of $\RR$ and whose topology is $\tau_\RR$:

$T_\RR := \struct {S / \RR, \tau_\RR}$

Also see