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

A topological space X is totally disconnected if the connected components in X are the one-point sets. Analogously, a topological space X is totally path-disconnected if all path-components in X 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 X is totally separated if for every x\in X, the intersection of all clopen neighborhoods of x is the singleton {x}. Equivalently, for each pair of distinct points x, y\in X, there is a pair of disjoint open neighborhoods U, V of x, y such that. Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take X to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then X 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 ) totally disconnected spaces are sometimes called hereditarily disconnected, while the terminology totally disconnected is used for totally separated spaces.

Examples

The following are examples of totally disconnected spaces:

Properties

Constructing a totally disconnected quotient space of any given space

Let X be an arbitrary topological space. Let x\sim y if and only if (where denotes the largest connected subset containing x). This is obviously an equivalence relation whose equivalence classes are the connected components of X. Endow X/{\sim} with the quotient topology, i.e. the finest topology making the map continuous. With a little bit of effort we can see that X/{\sim} 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 Y and any continuous map, there exists a unique continuous map with.

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