Tychonoff’s theorem for hyperspaces

Abstract:

If $\exp ({X_i})\backslash \{ \emptyset \}$ is equipped with a topology that preserves the topological convergence of nets of sets for every $i \in I$, then the Tychonoff product of the family $\{ \exp ({X_i})\backslash \{ \emptyset \} :i \in I\}$ is compact if and only if ${X_i}$ is compact for every $i \in I$. A similar result concerning sequential compactness is valid, for countable I.