Definition 3.9.4 (Quotient Space).label Let $X$ be a topological space and $\sim$ an equivalence relation on $X$, then there exists $(\til{X},\pi)$ such that
- (1)
$\til{X}$ is a topological space with underlying set $X/\sim$
- (2)
$\pi\in C(X;\til{X})$
- (3)
$\pi$ is constant on each equivalence class of $\sim$
- (U)
For any pair $(Y,f)$ satisfying (1),(2), and (3), there exists a unique $\til{f}\in C(\til{X};Y)$ such that the following diagram commutes
\[\xymatrix{ X \ar@{->}[r]|-{f} \ar@{->}[d]|-{\pi} & Y \\ \tilde{X} \ar@{->}[ru]|-{\tilde{f}} & }\] - (5)
$\pi$ is a quotient map
The space $(\til{X},\pi)$ is the quotient of $X$ by $\sim$.
Proof. Let $\til{X}=X/\sim$ and $\pi$ maps $x$ to the equivalence class containing $x$. For each $U\suf \til{X}$, define $U$ to be open if and only if $\pi\inv(U)\suf X$ is open, then $(\til{X},\pi)$ satisfies (1),(2),(3), and (5). For (U), since $f$ is constatn on each equivalence class of $\sim$, $\til{f}:\til{X}\to Y$ exists. For any $U\suf Y$ open, $\til{f}\inv(U)=\pi(f\inv(U))$ is saturated with respect to $\pi$, so $\til{f}\inv(U)$ is open.$\square$
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