Definition:Quotient Topology/Quotient Space
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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 $\RR$.
Let $\tau_\RR$ be the quotient topology on $S / \RR$ by $q_\RR$:
- $\tau_\RR := \set {U \subseteq S / \RR: \map {q_\RR^{-1} } U \in \tau}$
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 known as
A quotient space is also known as an identification space and a factor space.
However, note that an identification space is often (and on $\mathsf{Pr} \infty \mathsf{fWiki}$) used for a more general concept.
Also see
- Results about quotient spaces can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): quotient space