Definition

Quotient maps

1. Quotient map

   Let $X,Y$ be topological spaces and $p\colon X\to Y$ a surjective map. Then $p$ is called a quotient map if $U\subseteq Y$ is open if and only if $p^{-1}(U)$ is open in $X$.

2. Remarks

   • $p$ is a quotient map if and only if $p$ is surjective and $p^{-1}(A)$ is closed in $X$ if and only if $A\subseteq Y$ closed.
   • Maps which are open, continuous and surjective are quotient maps.
   • Maps which are closed, continuous and surjective are quotient maps.

Saturated subsets

1. Fiber

   Let $p\colon X\to Y$ be a surjective map between sets and $y\in Y$. The inverse image set

   $$p^{-1}(\{y\})=\{x\in X\mid p(x)=y\}$$

   will be denoted as $p^{-1}(y)$ and referred to as the fiber of $y$.

2. Saturated set

   Let $p\colon X\to Y$ be a surjective map between sets. We say that $A\subseteq X$ is saturated (with respect to $p$) if $A\cap p^{-1}(y)\ne\emptyset$ implies $p^{-1}(y)\subseteq A$

3. Remarks

   • $A$ is saturated if and only if $A=p^{-1}(p(A))$.
   • If $U\subseteq Y$, then $p^{-1}(U)$ is saturated.
   • $p$ is a quotient map if and only if $p$ is surjective, continuous, and $A$ saturated and open implies $p(A)$ open.
   • $p$ is a quotient map if and only if $p$ is surjective, continuous, and $A$ saturated and closed implies $p(A)$ closed.

Quotient topology

1. Quotient topology

   Let $X$ be a space, $A$ a set, and $p\colon X\to A$ a surjective map. Then there is exactly one topology on $A$ that makes $p$ a quotient map, such topology is called quotient topology.

2. Remark

   One must check that:

   $$\tau=\{U\subseteq A\mid \text{\(p^{-1}(U)\) is open in \(X\)}\}$$

   is a topology on $A$.

Partitions

1. Partition

   Let $X$ be a set. A partition $\tilde{X}$ of $X$ is a collection of nonempty disjoint subsets of $X$ with union $X$

2. Quotient space

   Let $X$ be a topological space, $\tilde{X}$ be a partition of $X$, and $p\colon X\to \tilde{X}$ the natural surjection. If we give $\tilde{X}$ the quotient topology induced by $p$, the space $\tilde{X}$ is called a quotient space of $X$

Quotients and subspaces

1. If $p\colon X\to Y$ is a quotient map and $A\subseteq X$ is a subspace, it does not necessarily follow that the restriction of $p$: $A\to p(A)$ is a quotient map. However we have:

2. Quotients and subspaces

   Let $p\colon X\to Y$ be a quotient map and $A\subseteq X$ a subspace, that is saturated with respect to $p$, and let $q\colon A\to p(A)$ the restriction of $p$. Then:

   • if $A$ is open or closed, then $q$ is a quotient map,
   • if $p$ is open or closed, then $q$ is a quotient map.

Proof

• We verify the following:

  $$\begin{align*} q^{-1}(V) &=p^{-1}(V) &\text{if \(V\subseteq p(A)\)},\\ p(U\cap A) &= p(U)\cap p(A) &\text{if \(U\subseteq X\)}. \end{align*}$$

Quotients and compositions

1. Teorema

   Composition of quotient maps is a quotient map.

Maps going out of quotients

Fundamental theorem

1. Fundamental theorem

   Let $p\colon X\to Y$ be a quotient map. Let $Z$ be a space, and $g\colon X\to Z$ a continuous map that is constant on each fiber of $p$, that is $p(x)=p(x')$ implies $g(x)=g(x')$. Then $g$ induces a continuous map $\overline{g}\colon Y\to Z$ such that $\overline{g}\circ p=g$. The function $\overline{g}$ is unique with the property that $\overline{g}\circ p=g$

   

Fundamental theorem

1. Fundamental theorem, for equivalence relations

   Let $X$ be a space, and $\sim$ an equivalence relation on $X$. Let $g\colon X\to Z$ be a continuous map such that $g(x)=g(x')$ whenever $x\sim x'$. Then, if we denote by $p\colon X\to X/\!\!\sim$ the quotient map, the map $g$ induces a unique continuous map $\overline{g}\colon X/\!\!\sim\,\to Z$ such that $\overline{g}\circ p=g$