Quotient Mapping/Examples/Modulo 2 pi as Angular Measurement

Example of Quotient Mapping

Let $\RR$ denote the congruence relation modulo $2 \pi$ on the real numbers $\R$ defined as:

$\forall x, y \in \R: \tuple {x, y} \in \RR \iff \text {$x$ and $y$}$ measure the same angle in radians

From Congruence Modulo $2 \pi$ as Angular Measurement, the quotient set induced by $\RR$ is:

$\R / \RR = \set {\eqclass \theta {2 \pi}: 0 \le \theta < 2 \pi}$

where:

$\eqclass \theta {2 \pi} = \set {\theta + 2 k \pi: k \in \Z}$

Hence the quotient mapping $q_\RR: \R \to \R / \RR$ is defined as:

$\forall x \in \R: \map {q_\RR} x = \eqclass x {2 \pi} = \set {x + 2 k \pi: k \in \Z}$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Quotient Functions