There's no reason for using \text, neither around \footnote nor inside it.
\documentclass{beamer}
\begin{document}
\begin{frame}
\begin{definition}[Clopen Set]
$A$ is a clopen set iff
\begin{equation}
A \in \mathcal{A}\mathrel{&}\neg A \in \mathcal{A}
\footnote{Equivalently, $A \in\mathcal{A} \cap \neg \mathcal{A}$}
\end{equation}
\end{definition}
\end{frame}
\end{document}
Note \mathrel{\&} in order to get better spacing.
But why a footnote?
\documentclass{beamer}
\begin{document}
\begin{frame}
\begin{definition}[Clopen Set]
$A$ is a clopen set iff
\begin{equation}
A \in \mathcal{A}\mathrel{&}\neg A \in \mathcal{A}
\footnote{Equivalently, $A \in\mathcal{A} \cap \neg \mathcal{A}$}
\end{equation}
\end{definition}
\end{frame}
\begin{frame}
\begin{definition}[Clopen Set]
$A$ is a clopen set iff
\begin{equation}
A \in \mathcal{A}\mathrel{&}\neg A \in \mathcal{A}
\end{equation}
Equivalently, $A \in\mathcal{A} \cap \neg \mathcal{A}$
\end{definition}
\end{frame}
\end{document}
