In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets.
In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the only connected subsets.
An important example of a totally disconnected space is the Cantor set, which is homeomorphic to the set of p-adic integers.
Another example, playing a key role in algebraic number theory, is the field Qp of p-adic numbers.
is totally disconnected if the connected components in
[1][2] Analogously, a topological space
is totally path-disconnected if all path-components in
Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents are singletons.
, the intersection of all clopen neighborhoods of
Equivalently, for each pair of distinct points
, there is a pair of disjoint open neighborhoods
Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces.
to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed.
is totally disconnected but its quasicomponents are not singletons.
For locally compact Hausdorff spaces the two notions (totally disconnected and totally separated) are equivalent.
Confusingly, in the literature (for instance[3]) totally disconnected spaces are sometimes called hereditarily disconnected,[4] while the terminology totally disconnected is used for totally separated spaces.
[4] The following are examples of totally disconnected spaces: Let
be an arbitrary topological space.
denotes the largest connected subset containing
In fact this space is not only some totally disconnected quotient but in a certain sense the biggest: The following universal property holds: For any totally disconnected space
, there exists a unique continuous map