Totally incoherent space

Totally incoherent spaces are examined in the mathematical sub-area of topology . In every topological space , one-element subsets and the empty set are connected . The totally disconnected spaces are characterized by the fact that there are no further connected subsets in them.

Probably the best known example is the Cantor crowd . Totally disjointed spaces appear in many mathematical theories.

definition

A topological space is called totally disconnected if there are no other connected subsets besides the empty and the one-element subsets.

Examples

• Discrete rooms , zero-dimensional rooms , totally separated rooms as well as extremely incoherent rooms are totally incoherent.
• ${\displaystyle \mathbb {Q} }$with the subspace topology of is totally incoherent. If there is a subset with at least two elements, there is an irrational number between them . The subspace is then the union of the two relatively open sets and therefore not contiguous. ${\displaystyle \mathbb {R} }$ ${\displaystyle X\subset \mathbb {Q} }$ ${\displaystyle a}$ ${\displaystyle X}$ ${\displaystyle \{x\in X;x<a\}}$ ${\displaystyle \{x\in X;x>a\}}$
• ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$with the subspace topology of is totally incoherent. ${\displaystyle \mathbb {R} }$
• The Cantor set is a totally incoherent compact Hausdorff space .
• The Baire room .
• The Troubled Frey Straight and the Troubled Frey Plane are totally disconnected.
• Pro-finite groups are precisely the totally disconnected compact topological groups .

properties

• Sub-spaces and products of totally disconnected spaces are totally disconnected again.
• Every continuous mapping from a coherent space into a totally disconnected space is constant, because the image is again coherent and therefore one-element.

Applications

Boolean algebras

According to Stone's theorem of representation , for every Boolean algebra there is a totally unrelated, compact Hausdorfraum , which is uniquely determined except for homeomorphism , so that the Boolean algebra is isomorphic to the algebra of the open-ended subsets of . Therefore, totally incoherent, compact Hausdorff rooms are also called Boolean rooms in this context . ${\displaystyle X}$ ${\displaystyle X}$

C * algebras

According to Gelfand-Neumark's theorem, every commutative C * -algebra is isometrically isomorphic to the algebra of continuous functions for a locally compact Hausdorff space that is uniquely determined except for homeomorphism . The following applies: ${\displaystyle A}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle X_{A}}$

• A commutative, separable C * -algebra is AF-C * -algebra if and only if is totally disconnected. ${\displaystyle X_{A}}$

p-adic numbers

The whole p-adic numbers for a prime number can be represented as series of the form with . With this you can identify with what makes a totally disjointed, compact Hausdorff area. Then the field of p-adic numbers is a σ-compact , locally compact, totally disconnected space. ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle p}$ ${\displaystyle \textstyle \sum _{i=0}^{\infty }a_{i}p^{i}}$ ${\displaystyle a_{i}\in \{0,\ldots ,p-1\}}$ ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle \{0,\ldots ,p-1\}^{\mathbb {N} _{0}}}$ ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle \textstyle \mathbb {Q} _{p}=\bigcup _{n=0}^{\infty }p^{-n}\mathbb {Z} _{p}}$

Individual evidence

1. ↑ Philip J. Higgins: An Introduction to Topological Groups (= London Mathematical Society Lecture Note Series. Vol. 15). Cambridge University Press, London et al. 1974 (recte 1975), ISBN 0-521-20527-1 , Chapter II.7, sentence 9.
2. ^ Paul R. Halmos : Lectures on Boolean Algebra. Springer, New York NY et al. 1974, ISBN 0-387-90094-2 , § 18, Theorem 6, Theorem 7.
3. ↑ Kenneth R. Davidson: C * -Algebras by Example (= Fields Institute Monographs. Vol. 6). American Mathematical Society, Providence RI 1996, ISBN 0-8218-0599-1 , Example III.2.5.