# What does it mean for the empty set to be connected and totally disconnected?

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.

But there are also some posts telling me that the empty set is totally disconnected. Like Here: http://forums.xkcd.com/viewtopic.php?f=17&t=19492 or here Is the Empty set an orientable manifold?

Can someone please elaborate what it means for the empty set is totally disconnected and does that imply that the empty set is disconnected?