I am trying to prove that the empty set is disconnected, but every single post I can find on this topic is about showing empty set is connected.
Recall definition of connected. A set $S$ is connected if the set is not the disjoint union of proper clopen subsets $A \sqcup B$.
Then $\varnothing$ is by definition connected, since $\varnothing$ is clopen and $S = \varnothing$ is again clopen. So the only clopen sets of $S = \varnothing$ are $S$ and $\varnothing$ and neither are proper.
Can someone please elaborate what it means for the empty set is totally disconnected and does that imply that the empty set is disconnected?