Totally disconnected space

Total disjointed spaces are investigated in the mathematical branch of topology. In any topological space one element subsets and the empty set are contiguous. The total incoherent spaces are characterized in that there are no other contiguous subsets within them.

The most famous example is the Cantor set. Total disjointed spaces occur in many mathematical theories.

Definition

A topological space is called totally disconnected if there are no other contiguous subsets beside the empty and singleton subsets.

Examples

• Discrete spaces, zero-dimensional spaces and extremal disconnected spaces are totally disconnected.
• With the subspace topology is totally disconnected. Indeed, if a subset with at least two elements, so there is between this an irrational number. The subspace is then the union of two relatively open sets and therefore will not contiguous.
• With the subspace topology is totally disconnected.
• The Cantor set is a totally disconnected compact Hausdorff space.
• The Baire space.
• The Sorgenfrey line of the Sorgenfrey level are totally disconnected.

Properties

• Subspaces and products totally disconnected spaces are totally disconnected again.
• Every continuous map from a connected space in a totally unrelated area is constant, because the image is re- connected and therefore a singleton.

Applications

Boolean algebras

According to the representation theorem of Stone there is any Boolean algebra one up to homeomorphism uniquely determined, totally disconnected, compact Hausdorrfraum so that the Boolean algebra isomorphic to the algebra of open - closed subsets of is .. Therefore, it is called totally disconnected, compact Hausdorff spaces in this context, Boolean spaces.

C *-algebras

Every commutative C * -algebra by the theorem of Gelfand - Neumark isometrically isomorphic to the algebra of continuous functions for one up to homeomorphism uniquely determined locally compact Hausdorff space. The following applies:

• A commutative, separable C * -algebra if and only AF-C *-algebra if it is totally disconnected.

P- adic numbers

All the p- adic numbers a prime number are known to be represented as a series of the form. Thus one can identify with what makes you a totally unrelated, compact Hausdorff space. Then the body of the p -adic numbers is a σ - compact, locally compact, totally disconnected space.