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 Qp of p-adic numbers.
Definition
A topological space [math]\displaystyle{ X }[/math] is totally disconnected if the connected components in [math]\displaystyle{ X }[/math] are the one-point sets.[1][2] Analogously, a topological space [math]\displaystyle{ X }[/math] is totally path-disconnected if all path-components in [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math] is totally separated space if and only if for every [math]\displaystyle{ x\in X }[/math], the intersection of all clopen neighborhoods of [math]\displaystyle{ x }[/math] is the singleton [math]\displaystyle{ \{x\} }[/math]. Equivalently, for each pair of distinct points [math]\displaystyle{ x, y\in X }[/math], there is a pair of disjoint open neighborhoods [math]\displaystyle{ U, V }[/math] of [math]\displaystyle{ x, y }[/math] such that [math]\displaystyle{ X= U\sqcup V }[/math].
Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take [math]\displaystyle{ X }[/math] to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then [math]\displaystyle{ X }[/math] is totally disconnected but its quasicomponents are not singletons. For locally compact Hausdorff spaces the two notions (totally disconnected and totally separated) are equivalent.
Unfortunately in the literature (for instance [3]), 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:
- 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[math]\displaystyle{ \, \cap \, \mathbb{Q}^{\omega} }[/math] 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 [math]\displaystyle{ X }[/math] be an arbitrary topological space. Let [math]\displaystyle{ x\sim y }[/math] if and only if [math]\displaystyle{ y\in \mathrm{conn}(x) }[/math] (where [math]\displaystyle{ \mathrm{conn}(x) }[/math] denotes the largest connected subset containing [math]\displaystyle{ x }[/math]). This is obviously an equivalence relation whose equivalence classes are the connected components of [math]\displaystyle{ X }[/math]. Endow [math]\displaystyle{ X/{\sim} }[/math] with the quotient topology, i.e. the finest topology making the map [math]\displaystyle{ m:x\mapsto \mathrm{conn}(x) }[/math] continuous. With a little bit of effort we can see that [math]\displaystyle{ X/{\sim} }[/math] 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 [math]\displaystyle{ Y }[/math] and any continuous map [math]\displaystyle{ f : X\rightarrow Y }[/math], there exists a unique continuous map [math]\displaystyle{ \breve{f}:(X/\sim)\rightarrow Y }[/math] with [math]\displaystyle{ f=\breve{f}\circ m }[/math].
See also
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.
References
- Willard, Stephen (2004), General topology, Dover Publications, ISBN 978-0-486-43479-7 (reprint of the 1970 original, MR0264581)
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