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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?

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Recall the definitions:

A space is connected if it cannot be written as union of two non-empty disjoint open subsets.

A space is disconnected if it is not connected.

A space is totally disconnected if it has no non-trivial connected subsets.

What about the empty space? It is connected, in fact vacuously so as it lacks non-empty subsets in the first place. Consequently it is not disconnected. On the other hand it is totally disconnected as its only subsets are (connected but) trivial.

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