Union of Topologies on Singleton or Doubleton is Topology

Theorem
Let $S$ be a set containing either exactly one or exactly two elements.

Let $\left({\tau_i}\right)_{i \in I}$ be an arbitrary indexed set of topologies for a set $S$.

Then $\tau := \displaystyle \bigcup_{i \in I} {\tau_i}$ is also a topology for $X$.