商映射$f,f'$,双射$ϕ$使$ϕ∘f=f'$

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在Proposition 3.9,3.35和MSE上这个回答都提到:

a continuous surjection $f:X→Y$ is a quotient map if and only if the following is satisfied for all functions $g:Y→Z$:
$g$ is continuous if and only if $g∘f:X→Z$ is continuous.

它的推论

Given two quotient maps $f:X→Y,f':X→Y'$ and a bijection $ϕ:Y→Y'$ such that $ϕ∘f=f'$. Then $ϕ$ is a homeomorphism.

证明:
在上面定理中,取$g=ϕ$,因为$f$是商映射且$ϕ∘f=f'$连续,所以$ϕ$连续.
取$g=ϕ^{-1}$,因为$f^{-1}$是商映射且$ϕ∘f'=f$连续,所以$ϕ^{-1}$连续.
所以$ϕ$是同胚.