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 proper 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.

DefinitionEdit

A topological space   is totally disconnected if the connected components in   are the one-point sets. Analogously, a topological space   is totally path-disconnected if all path-components in   are the one-point sets.


Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents are singletons. Equivalently, a topological space   is totally separated space if and only if for every  , the intersection of all clopen neighborhoods of   is the singleton  . Equivalently, for each pair of distinct points  , there is a pair of disjoint open neighborhoods   of   such that  .

Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take   to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then   is totally disconnected but its quasicomponents are not singletons. For locally compact Hausdorff spaces the two notions (totally disconnected and totally separated) are equivalent.


Unfortunately in the literature (for instance [1]), totally disconnected spaces are sometimes called hereditarily disconnected while the terminology totally disconnected is used for totally separated spaces.



ExamplesEdit

The following are examples of totally disconnected spaces:

PropertiesEdit

Constructing a totally 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

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

See alsoEdit

  1. ^ Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Sigma Series in Pure Mathematics. ISBN 3-88538-006-4.