and related branches of mathematics
, a totally disconnected space
is a topological space
which is maximally disconnected, in the sense that it has no non-trivial connected
subsets. In every topological space the empty set and the one-point sets are connected; in a totally disconnected space these are the only
An important example of a totally disconnected space is the Cantor set. Another example, playing a key role in algebraic number theory, is the field Qp of p-adic numbers.
A topological space X
is totally disconnected
if the connected components
are the one-point sets.
The following are examples of totally disconnected spaces: