Wronskian

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 Q_p 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:

• Discrete spaces
• The rational numbers
• The irrational numbers
• The p-adic numbers; more generally, profinite groups are totally disconnected.
• The Cantor set
• The Baire space
• The Sorgenfrey line
• Zero dimensional T₁ spaces
• Extremally disconnected Hausdorff spaces
• Stone spaces
• The Knaster–Kuratowski fan provides an example of a connected space, such that the removal of a single point produces a totally disconnected space.
• The Erdős space ℓ^p(Z)∩${\mathbb{\mathbb{Q}}}^{\omega }$ is a totally disconnected space that does not have dimension zero.

Properties

• Subspaces, products, and coproducts of totally disconnected spaces are totally disconnected.
• Totally disconnected spaces are T₁ spaces, since points are closed.
• Continuous images of totally disconnected spaces are not necessarily totally disconnected, in fact, every compact metric space is a continuous image of the Cantor set.
• A locally compact hausdorff space is zero-dimensional if and only if it is totally disconnected.
• Every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces.
• It is in general not true that every open set is also closed.
• It is in general not true that the closure of every open set is open, i.e. not every totally disconnected Hausdorff space is extremally disconnected.

Constructing a disconnected space

Let $X$ be an arbitrary topological space. Let $x\sim y$ if and only if $y\in \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}(x)$ (where $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}(x)$ 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 $m:x\mapsto \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}(x)$ continuous. With a little bit of effort we can see that $X/\sim$ is totally disconnected. We also have the following universal property: if $f:X\to Y$ a continuous map to a totally disconnected space, then it uniquely factors into $f=\breve{f}\circ m$ where $\breve{f}:(X/\sim )\to Y$ is continuous.

References

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See also

• Totally disconnected groups.