Totally disconnected

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 X be an arbitrary topological space. Let x\sim y if and only if y\in \mathrm{conn}(x) (where \mathrm{conn}(x) denotes the largest connected subset containing x). This is obviously an equivalence relation. Endow X/{\sim} with the quotient topology, i.e. the coarsest topology making the map m:x\mapsto \mathrm{conn}(x) continuous. With a little bit of effort we can see that X/{\sim} is totally disconnected. We also have the following universal property: if f : X\rightarrow Y a continuous map to a totally disconnected space, then it uniquely factors into f=\breve{f}\circ m where \breve{f}:(X/\sim)\rightarrow Y is continuous.

References

  • (reprint of the 1970 original, 0264581)

See also

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