Totally disconnected space

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 (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. 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 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.


The following are examples of totally disconnected spaces:


Constructing a totally disconnected space

Let be an arbitrary topological space. Let if and only if (where denotes the largest connected subset containing ). 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 .


  • Willard, Stephen (2004), General topology, Dover Publications, ISBN 978-0-486-43479-7, MR 2048350 (reprint of the 1970 original, MR0264581)

See also

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