Theorem 9.1.
Let \(X^*\) be a partition of a space \(X\) with topology \(\mc T\text{.}\) Then the set \(\mc T^*=\setB{U^*\subseteq X^*}{\bigcup U^*\in\mc T}\) defines a topology on \(X^*\text{.}\)
Section 9 Quotient Spaces
Definition 9.2.
The topology defined in Theorem 9.1 is called the quotent topology, and \(X^*\) is called a quotient space or identification space.
Theorem 9.3.
Let \((X\times Y)^*=\setBuilder{\setList{x}\times Y}{x\in X}\) partition the product \(X\times Y\text{.}\) Then the quotient space \((X\times Y)^*\) is homeomorphic to \(X\text{.}\)
Definition 9.4.
A subset \(R\subseteq X^2\) is called a relation on \(X\text{,}\) where the notation \(xRy\) is equivalent to writing \(\tuple{x,y}\in R\text{.}\)
A relation \(\sim\) on \(X\) is called an equivalence relation if it satisfies the following for all \(x,y,z\in X\text{.}\)
- \(x\sim x\text{.}\) (Reflexivity)
- \(x\sim y\) implies \(y\sim x\text{.}\) (Symmetry)
- \(x\sim y\) and \(y\sim z\) implies \(y\sim z\text{.}\) (Transitivity)
Proposition 9.5.
Let \(X^*\) be a partition of \(X\) and define the relation \(\sim\) on \(X\) such that \(x\sim y\) if and only if \(\setList{x,y}\subseteq A\) for some \(A\in X^*\text{.}\) Then \(\sim\) is an equivalence relation.
Theorem 9.6.
Let \(\sim\) be an equivalence relation on \(X\text{,}\) and let \([x]=\setBuilder{y\in X}{x\sim y}\text{.}\) Then \(X^*=\setBuilder{[x]}{x\in X}\) is a partition of \(X\text{.}\)
Definition 9.7.
Let \(\sim\) be an equivalence relation on a topological space \(X\text{.}\) Then \(X/\sim\) denotes the quotient space defined by the partition \(X^*\) given in Theorem 9.6.
Proposition 9.8.
Let \(R\) be a relation on \(X\text{.}\) Then
\begin{equation*}
\sim=\bigcap\setBuilder{\sim^\star\subseteq X^2}{R\subseteq\sim^\star\text{ and }\sim^\star
\text{ is an equivalence relation on }X}
\end{equation*}
is an equivalence relation on \(X\text{.}\)
(Therefore, an equivalence relation may be defined as the minimal equivalence relation satisfying a list of relationships.)
Proposition 9.9.
Define the equivalence relation \(\sim_Y\) on \(X\times Y\) by \(\tuple{x,a}\sim_Y\tuple{x,b}\text{.}\) Then the quotient space \((X\times Y)/\sim_Y\) is homeomorphic to \(X\text{.}\)
Checkpoint 9.10. Curves and surfaces defined as quotients.
Show that each of the following Euclidean subspaces and quotients of Euclidean subspaces are homeomorphic.
- \(X=[0,1]/\sim\) where \(0\sim 1\text{,}\) and \(Y=\setBuilder{\tuple{x,y}\in\mb R^2}{x^2+y^2=1}\text{.}\)
- \(X=[0,2]/\sim\) where \(0\sim 1\sim 2\text{,}\) and \(Y=\setBuilder{\tuple{x,y}\in\mb R^2}{(x-1)^2+y^2=1}\cup\setBuilder{\tuple{x,y}\in\mb R^2}{(x+1)^2+y^2=1}\text{.}\)
- \(X=[0,1]^2/\sim\) where \(\tuple{0,y}\sim\tuple{1,y}\text{,}\) and \(Y=\setBuilder{\tuple{x,y}\in\mb R^2}{1\leq x^2+y^2\leq 2}\text{.}\)
- \(X=[0,1]^2/\sim\) where \(\tuple{x,y}\sim\tuple{z,w}\) whenever at least one of \(x,y\) and at least one of \(z,w\) is in \(\setList{0,1}\text{,}\) and \(Y=\setBuilder{\tuple{x,y,z}\in\mb R^2}{1\leq x^2+y^2+z^2=1}\text{.}\)
Definition 9.11.
The hypersphere of dimension \(n\) is the quotient space \(S^n=[0,1]^n/\sim\) given by \(\vec x\sim\vec y\) whenever there exist \(i,j\in\setList{0,\dots,n}\) such that \(\vec x(i),\vec y(j)\in\setList{0,1}\text{.}\)
Definition 9.12.
The Möbius strip is the quotient space \(M=[0,1]^2/\sim\) given by \(\tuple{0,y}\sim\tuple{1,1-y}\text{.}\)
Definition 9.13.
The torus is the quotient space \(T=[0,1]^2/\sim\) given by \(\tuple{0,y}\sim\tuple{1,y}\) and \(\tuple{x,0}\sim\tuple{x,1}\text{.}\)
Definition 9.14.
The Klein bottle is the quotient space \(K=[0,1]^2/\sim\) given by \(\tuple{0,y}\sim\tuple{1,1-y}\) and \(\tuple{x,0}\sim\tuple{x,1}\text{.}\)
