Definition:Quotient Topology
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Definition
Let $T = \struct {S, \tau}$ 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 $\RR$.
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
- Results about the quotient topology can be found here.
Linguistic Note
The word quotient derives from the Latin word meaning how often.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): quotient topology