Definition 3.9.2 (Quotient Map).label Let $X$ be a topological space, $Y$ a set and $\pi:X\to Y$ a surjection then the following are equivalent:
- (1)
$Y$ has the final topology generated by $\pi$
- (2)
$\pi\in C(X;Y)$ and for any saturated open set $U\suf X$, $\pi(U)$ is open
- (U)
For each topological space $Z$ and function $f:Y\to Z$, $f\in C(Y;Z)\iff f\circ\pi\in C(X;Z)$
If the above holds, then $\pi$ is a quotient map.
Proof. Suppose (1) holds then (2) follows from Definition 3.1.8 and Definition 3.9.1. Suppose (2) holds then by continuity of $\pi$, $\pi\inv(V)$ is open whenever $V\suf Y$ is open. Observe $V=\pi(\pi\inv(V))$ for any $V\suf Y$ since $\pi$ is surjective hence $\pi\inv(V)=\pi\inv(\pi(\pi\inv(V)))$ so if $\pi\inv(V)$ is open then by assumption $\pi(\pi\inv(V))$ is open, showing (1). Suppose (1) holds then $f\circ \pi$ is continuous whenever $f$ is continuous. If $f\circ\pi$ is continuous then $f$ is continuous by Definition 3.1.8(U’), showing (U). Suppose (U) holds then Definition 3.1.8(4) shows (1).$\square$
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