Definition:Totally Disconnected Space

Definition
A topological space is called totally disconnected if all connected components are singletons.

That is, a topological space is totally disconnected if it contains no non-degenerate connected sets, i.e. connected sets containing more than one point.

Remark
Note that a topological space containing at most one element is both connected and totally disconnected.

Hence, the definition is sometimes applied only to spaces having at least two elements.

Also see

 * Definition:Totally Pathwise Disconnected Space