Quotient Map

Definition

A function $p:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$ between two topological spaces is a quotient map if it satisfies the following conditions:

• $p$ is Surjective
• $U\in {\mathcal{T}}_Y\iff p^{-1}(U)\in {\mathcal{T}}_X\text{or}(Y\setminus U)\in {\mathcal{T}}_Y\iff (X\setminus p^{-1}(U))\in {\mathcal{T}}_X$
  • Equivalent: $p\in C^0$ ($p$ is continuous) and $p$ is open or Closed Map.

Facts

Consider two quotient maps $p:X\to Y$ and $f:Y\to Z$. The composite map of the quotient maps $f\circ p$ is a quotient map

Consider a quotient map $p:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$ between two topological spaces, a subset $A\subset X\text{s.t.}{p}^{-1}(p(A))$, and a function $p{|}_A:A\to p(A)$ with restricted domain. Then,

• If $A$ is either open or closed in $X$, then the function $p{|}_A$ is a quotient map $A\in {\mathcal{T}}_{\mathcal{X}}\text{or}(X\setminus A)\in {\mathcal{T}}_X⟹p{|}_A\text{is a quotient map}$
• If $p$ is either an Open Map or a Closed Map, then the function $p{|}_A$ is a quotient map.

Consider a quotient map $p:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$, and a function $g:(X,{\mathcal{T}}_X)\to (Z,{\mathcal{T}}_Z)$ satisfying $\forall y\in Y,g{|}_{p^{-1}}(\{y\})=c$ where $c$ is a constant, between topological spaces. Then,

• There exists a function such that the composite function $f\circ p$ is equal to $g$ $\exists f:(Y,{\mathcal{T}}_Y)\to (Z,{\mathcal{T}}_Z)\text{s.t.}f\circ p=g$
• $f$ is continuous if and only if $g$ is continuous $f\in C^0\iff g\in C^0$
• $f$ is a quotient map $\iff$ $g$ is a quotient map

Consider a Surjective continuous map $g:(X,{\mathcal{T}}_X)\to (Z,{\mathcal{T}}_Z)$ between topological spaces, and a Partition $X^{\ast }=\{g^{-1}(\{z\})|z\in Z\}$. We can generate a Quotient Topology ${\mathcal{T}}_q$ with the map $q$ that maps each point $x\in X$ to its Equivalence Class $[x]$ in $X^{\ast }$. Then,

• There exists a Bijective continuous map $f:X^{\ast }\to Z$ and $f$ is a Homeomorphism if and only if $g$ is a quotient map.
• If $Z$ is Hausdorff Space then $X^{\ast }$ is Hausdorff Space.