Totally disconnected space

Not to be confused with extremally 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 ${\displaystyle X}$ is totally disconnected if the connected components in ${\displaystyle X}$ are the one-point sets. ^{[1] [2]} Analogously, a topological space ${\displaystyle X}$ is totally path-disconnected if all path-components in ${\displaystyle X}$ 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 ${\displaystyle X}$ is totally separated if for every ${\displaystyle x\in X}$, the intersection of all clopen neighborhoods of ${\displaystyle x}$ is the singleton ${\displaystyle \{x\}}$. Equivalently, for each pair of distinct points ${\displaystyle x,y\in X}$, there is a pair of disjoint open neighborhoods ${\displaystyle U,V}$ of ${\displaystyle x,y}$ such that ${\displaystyle X=U\sqcup V}$.

Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take ${\displaystyle X}$ to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then ${\displaystyle X}$ 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${\displaystyle \,\cap \,\mathbb {Q} ^{\omega }}$ 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 ${\displaystyle X}$ be an arbitrary topological space. Let ${\displaystyle x\sim y}$ if and only if ${\displaystyle y\in \mathrm {conn} (x)}$ (where ${\displaystyle \mathrm {conn} (x)}$ denotes the largest connected subset containing ${\displaystyle x}$). This is obviously an equivalence relation whose equivalence classes are the connected components of ${\displaystyle X}$. Endow ${\displaystyle X/{\sim }}$ with the quotient topology, i.e. the finest topology making the map ${\displaystyle m:x\mapsto \mathrm {conn} (x)}$ continuous. With a little bit of effort we can see that ${\displaystyle X/{\sim }}$ 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 ${\displaystyle Y}$ and any continuous map ${\displaystyle f:X\rightarrow Y}$, there exists a unique continuous map ${\displaystyle {\breve {f}}:(X/\sim )\rightarrow Y}$ with ${\displaystyle f={\breve {f}}\circ m}$.

See also

• Extremally disconnected space
• Totally disconnected group

Citations

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. 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, MR 0264581)
• Kuratowski, Kazimierz (1968), Topology II: Transl. from French (Revised ed.), New York: Academic Press [u.a.], ISBN   9780124292024