Let $\mathcal{C}$ be a class whose elements are topological spaces. Define a relation $\sim$ on $\mathcal{C}$ by declaring that, for $(X,\tau_X),(Y,\tau_Y) \in \mathcal{C}$,
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\begin{align*}
(X,\tau_X) \sim (Y,\tau_Y) \iff \text{there exists a homeomorphism } f:X \to Y.
\end{align*}
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Then $\sim$ is an [equivalence relation](/page/Equivalence%20Relation) on $\mathcal{C}$.