In topology and related branches of mathematics, a totally disconnected space is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every topological space, the singletons 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. Another example, playing a key role in algebraic number theory, is the field of p-adic numbers.
Definition
A topological space X is totally disconnected if the connected components in X are the one-point sets. Analogously, a topological space X is totally path-disconnected if all path-components in X are the one-point sets.
Examples
The following are examples of totally disconnected spaces:
Subspaces, products, and coproducts of totally disconnected spaces are totally disconnected.
Totally disconnected spaces are T1 spaces, since singletons are closed.
Continuous images of totally disconnected spaces are not necessarily totally disconnected, in fact, every compact metric space is a continuous image of the Cantor set.
It is in general not true that every open set in a totally disconnected space is also closed.
It is in general not true that the closure of every open set in a totally disconnected space is open, i.e. not every totally disconnected Hausdorff space is extremally disconnected.
Constructing a totally disconnected space
Let be an arbitrary topological space. Let if and only if . This is obviously an equivalence relation whose equivalence classes are the connected components of. Endow with the quotient topology, i.e. the finest topology making the map continuous. With a little bit of effort we can see that is totally disconnected. We also have the following universal property: if a continuous map to a totally disconnected space, then there exists a unique continuous map with.
