Definition:Quotient Space

Definition

Let $X$ be a space with topology $\vartheta$ and an equivalence relation $\sim$ on $X$.

Then the quotient space $X/ \sim$ is defined as the space whose points are equivalence classes of $\sim$ (called the quotient set) and whose topology $\vartheta_{X/ \sim}$ is defined as:

$U \in \vartheta_{X/ \sim} \iff \pi^{-1} \left({U}\right) \in \vartheta$

where $\pi: X \to X/ \sim$ is the mapping which takes a point in $X$ to its equivalence class, called the quotient mapping.

Also see

• Quotient Topology

Sources

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