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.

Definition

A topological space X is totally disconnected if the connected components in X are the one-point sets.

Examples

The following are examples of totally disconnected spaces:

Properties

Constructing a 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. Endow with the quotient topology, i.e. the coarsest 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 it uniquely factors into where is continuous.

References

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See also