Totally disconnected space

In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as 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, which is homeomorphic to the set of p-adic integers. Another example, playing a key role in algebraic number theory, is the field Qp of p-adic numbers.

Definition

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A topological space   is totally disconnected if the connected components in   are the one-point sets.[1][2] 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. That is, a topological space   is totally separated 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.

Confusingly, in the literature (for instance[3]) totally disconnected spaces are sometimes called hereditarily disconnected,[4] while the terminology totally disconnected is used for totally separated spaces.[4]

Examples

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The following are examples of totally disconnected spaces:

Properties

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Constructing a totally disconnected quotient space of any given space

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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.

In fact this space is not only some totally disconnected quotient but in a certain sense the biggest: The following universal property holds: For any totally disconnected space   and any continuous map  , there exists a unique continuous map   with  .

See also

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Citations

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  1. ^ Rudin 1991, p. 395 Appendix A7.
  2. ^ Munkres 2000, pp. 152.
  3. ^ Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Sigma Series in Pure Mathematics. ISBN 3-88538-006-4.
  4. ^ a b Kuratowski 1968, pp. 151.

References

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