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Let $A$ and $C$ be two topological spaces. Let $B$, a subspace of $A$, and $D$, a subspace of $C$, be homeomorphic topological spaces. Suppose we also know that $A/B$ is homeomorphic to $C/D$. Does it follow that $A$ and $C$ are homeomorphic?

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No; for example, let $$\begin{align*}A&=\mathbb{S}^1=\{(x,y)\in\mathbb{R}^2\mid x^2+y^2=1\} \\\\ B&=\{(1,0),(-1,0)\}\\\\ C&=\mathbb{S}^1\cup\{(x,0)\in\mathbb{R}^2\mid x\in[-2,-1]\}\\\\ D&=\{(-2,0),(-1,0)\}\end{align*}$$ Then $A/B$ and $C/D$ are both homeomorphic to $\mathbb{S}^1\vee\mathbb{S}^1$, and $B$ and $D$ are homeomorphic, but $A$ and $C$ are not homeomorphic.

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