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Concept
Totally disconnected space
Summary
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.
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. That is, 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 ), totally disconnected spaces are sometimes called hereditarily disconnected, while the terminology totally disconnected is used for totally separated spaces.
The following are examples of totally disconnected spaces:
Discrete spaces
The rational numbers
The irrational numbers
The p-adic numbers; more generally, all profinite groups are totally disconnected.
Official source
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