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Totally disconnected space

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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 empty set and the one-point sets 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.

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DefinitionEdit

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.

ExamplesEdit

The following are examples of totally disconnected spaces:

PropertiesEdit

Constructing a disconnected spaceEdit

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  .

ReferencesEdit

See alsoEdit