Totally disconnected space
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 singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the only connected proper subsets.
A topological space is totally disconnected if the connected components in are the one-point sets. Analogously, a topological space is totally path-disconnected if all path-components in are the one-point sets.
Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents are singletons. Equivalently, a topological space is totally separated space if and only if for every , the intersection of all clopen neighborhoods of is the singleton . Equivalently, for each pair of distinct points , there is a pair of disjoint open neighborhoods of such that .
Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then 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 ), totally disconnected spaces are sometimes called hereditarily disconnected while the terminology totally disconnected is used for totally separated spaces.
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 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.
- 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 spaceEdit
Let be an arbitrary topological space. Let if and only if (where denotes the largest connected subset containing ). This is obviously an equivalence relation whose equivalence classes are the connected components of . Endow with the quotient topology, i.e. the finest topology making the map continuous. With a little bit of effort we can see that is totally disconnected. We also have the following universal property: if a continuous map to a totally disconnected space , then there exists a unique continuous map with .