Quotient Space

Definition

Consider a Topological Space $(X,\mathcal{T})$ and a Partition $X^{\ast }$ on it, and a Surjective map $p:X\to X^{\ast }$ maps each point $x\in X$ to its Equivalence Class $[x]$ in $X^{\ast }$ (element of $X^{\ast }$ containing it). Then, the set $X^{\ast }$ equipped with the Quotient Topology ${\mathcal{T}}_q$ generated by $p$ is the quotient space $(X^{\ast },{\mathcal{T}}_q)$ of $X$, and the map $p$ become a Quotient Map.

Examples

Consider a Standard Topology on Real Number $(\mathbb{R},\mathcal{T})$ and a quotient map $p:\mathbb{R}\to \{a,b,c\},p(x)=\{\begin{array}{ll}a& x>0\\ b& x<0\\ c& x=0\end{array}$. The quotient topology ${\mathcal{T}}_q$ on the set $\{a,b,c\}$ induced by $p$ is generated as ${\mathcal{T}}_q=\{\varnothing ,a,b,\{a,b\},\{a,b,c\}\}$

Consider a Standard Topology on real plane $({\mathbb{R}}^2,\mathcal{T})$ and a unit disk $X=D^2=\{(x,y)|x^2+y^2\le 1\}$ on it. The Quotient Topology constructed by a partition $X^{\ast }=\{(x,y)|x^2+y^2<1\}\cup \{S^1\}$ where $S^1=\{(x,y)|x^2+y^2=1\}$, forms a sphere.

Consider a Standard Topology on real plane $({\mathbb{R}}^2,\mathcal{T})$ and a rectangle $X=\{(x,y)|0\le x\le 1,0\le y\le 1\}$ on it. Consider a partition $X^{\ast }=\{(x,y)|0<x<1,0<y<1\}\cup \{(x,0),(x,1)|0<x<1\}\cup \{(0,y),(1,y)|0<y<1\}\cup \{(0,0)=(0,1)=(1,0)=(1,1)\}$ The Quotient Topology constructed by the partition forms a torus.