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 $Q p$ of p-adic numbers.

Definition

A topological space $X$ is totally disconnected if the connected components in $X$ are the one-point sets.^[1]^[2] 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 $X=U\sqcup V$ .

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^[3]) totally disconnected spaces are sometimes called hereditarily disconnected,^[4] while the terminology totally disconnected is used for totally separated spaces.^[4]

Examples

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.
The Cantor set and the Cantor space
The Baire space
The Sorgenfrey line
Every Hausdorff space of small inductive dimension 0 is totally disconnected
The Erdős space ℓ² $\,\cap \,\mathbb {Q} ^{\omega }$ is a totally disconnected Hausdorff space that does not have small inductive dimension 0.
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.

Properties

Subspaces, products, and coproducts of totally disconnected spaces are totally disconnected.
Totally disconnected spaces are T₁ spaces, since singletons 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 has small inductive dimension 0 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 in a totally disconnected space is also closed.
It is in general not true that the closure of every open set in a totally disconnected space is open, i.e. not every totally disconnected Hausdorff space is extremally disconnected.

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 $y\in \mathrm {conn} (x)$ (where $\mathrm {conn} (x)$ 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 $m:x\mapsto \mathrm {conn} (x)$ 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 $f:X\rightarrow Y$ , there exists a unique continuous map ${\breve {f}}:(X/\sim )\rightarrow Y$ with $f={\breve {f}}\circ m$ .

Citations

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

References

Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
Willard, Stephen (2004), General topology, Dover Publications, ISBN 978-0-486-43479-7, MR 2048350 (reprint of the 1970 original, MR0264581)
Kuratowski, Kazimierz (1968), Topology II: Transl. from French (Revised ed.), New York: Academic Press [u.a.], ISBN 9780124292024

[FOOTNOTERudin1991395_Appendix_A7-1] Rudin 1991, p. 395 Appendix A7.

[FOOTNOTEMunkres2000152-2] Munkres 2000, pp. 152.

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

[FOOTNOTEKuratowski1968151-4] Kuratowski 1968, pp. 151.

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