Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces, let $f: X \to Y$ be a homeomorphism, let $(I,\preceq)$ be a directed set, and let $(x_i)_{i \in I}$ be a net in $X$. For every $x \in X$,
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\begin{align*}
x_i \to x \text{ in } X \iff f(x_i) \to f(x) \text{ in } Y.
\end{align*}