Totally disconnected

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 {\displaystyle X} is totally disconnected if the connected components in X {\displaystyle X} are the one-point sets.^[1][2] Analogously, a topological space X {\displaystyle X} is totally path-disconnected if all path-components in X {\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 X {\displaystyle X} is totally separated if for every x ∈ X {\displaystyle x\in X} , the intersection of all clopen neighborhoods of x {\displaystyle x} is the singleton { x } {\displaystyle \{x\}} . Equivalently, for each pair of distinct points x , y ∈ X {\displaystyle x,y\in X} , there is a pair of disjoint open neighborhoods U , V {\displaystyle U,V} of x , y {\displaystyle x,y} such that X = U ⊔ V {\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 X {\displaystyle X} to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then X {\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^[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 ∩ Q ω {\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 connected components 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 {\displaystyle X} be an arbitrary topological space. Let x ∼ y {\displaystyle x\sim y} if and only if y ∈ c o n n ( x ) {\displaystyle y\in \mathrm {conn} (x)} (where c o n n ( x ) {\displaystyle \mathrm {conn} (x)} denotes the largest connected subset containing x {\displaystyle x} ). This is obviously an equivalence relation whose equivalence classes are the connected components of X {\displaystyle X} . Endow X / ∼ {\displaystyle X/{\sim }} with the quotient topology, i.e. the finest topology making the map m : x ↦ c o n n ( x ) {\displaystyle m:x\mapsto \mathrm {conn} (x)} continuous. With a little bit of effort we can see that X / ∼ {\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 Y {\displaystyle Y} and any continuous map f : X → Y {\displaystyle f:X\rightarrow Y} , there exists a unique continuous map f ˘ : ( X / ∼ ) → Y {\displaystyle {\breve {f}}:(X/\sim )\rightarrow Y} with f = f ˘ ∘ m {\displaystyle f={\breve {f}}\circ m} .

See also

• Extremally disconnected space
• Totally disconnected group

Citations

References

• Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. (accessible to patrons with print disabilities)
• 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