Definition:Totally Disconnected Space

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

Equivalently, a space is totally disconnected if it contains no nondegenerate connected subsets (i.e., connected subsets 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.