Extremely incoherent space

Extremely incoherent spaces are examined in the mathematical sub-area of topology (mathematics) . As the name suggests, these spaces are very far from coherent properties. They occur in the theory of Boolean algebras and the Abelian Von Neumann algebras .

definition

A topological space has bad connected properties if there are many open-closed subsets in it. If one demands in a T ₁ space that every closed set is already open, the space is discrete . A slightly weakened requirement leads to the term considered here:

A topological space is called extremally incoherent if the closure of every open set is open again.

Examples

Discrete spaces are extremely disjointed. In metric spaces, the reverse applies, that is, extremely disjointed, metrizable spaces are discrete.
The Stone-Čech compactification of natural numbers is a non-discrete, extremely disjointed space. ${\ displaystyle \ beta (\ mathbb {N})}$

properties

Extremely disjointed spaces are totally disjointed . The reverse is not true, as the example of the Cantor set shows.
Extremely incoherent Hausdorff areas are even totally separated ; the reverse does not apply here either.
For every Borel set of an extremely disconnected compact Hausdorff space there is a unique open-closed set , so that the symmetrical difference is a lean set . ${\ displaystyle B}$ ${\ displaystyle A}$ ${\ displaystyle A \, \ triangle \, B}$

If it is an open and dense subset of an extremely disconnected compact Hausdorff space and is a continuous and bounded function, there is a uniquely determined continuous function that continues. ${\ displaystyle U}$ ${\ displaystyle f \ colon U \ rightarrow \ mathbb {C}}$ ${\ displaystyle {\ tilde {f}} \ colon X \ rightarrow \ mathbb {C}}$ ${\ displaystyle f}$

Applications

Boolean algebras

According to Stone's theorem of representation, there is a totally incoherent space for every Boolean algebra , the so-called Boolean space for algebra, so that the algebra is isomorphic to the set algebra of the open-closed sets in . The following applies: ${\ displaystyle X}$ ${\ displaystyle X}$

A Boolean algebra is complete (that is, every bounded set has a supremum and an infimum) if and only if the associated Boolean space is extremely disconnected.

Abelian Von Neumann algebras

Abelian Von Neumann algebras are, as C * algebra according to the Gelfand-Neumark theorem, isometrically isomorphic to the algebra of continuous functions for a compact Hausdorff space that is uniquely determined except for homeomorphism . For abelian Von Neumann algebras is extremal disconnected. ${\ displaystyle C (X)}$ ${\ displaystyle X \ rightarrow \ mathbb {C}}$ ${\ displaystyle X}$ ${\ displaystyle X}$

The converse does not apply, that is, there are extremely disconnected, compact Hausdorff spaces , so that the algebra is not isomorphic to a Von Neumann algebra. ${\ displaystyle X}$ ${\ displaystyle C (X)}$

Individual evidence

^ Richard V. Kadison , John R. Ringrose : Fundamentals of the Theory of Operator Algebras. Special Topics. Volume 1: Elementary Theory (= Pure and Applied Mathematics. Vol. 100, 1). Academic Press, Boston MA et al. 1983, ISBN 0-12-393301-3 , 5.2.10.
^ Richard V. Kadison, John R. Ringrose: Fundamentals of the Theory of Operator Algebras. Special Topics. Volume 1: Elementary Theory (= Pure and Applied Mathematics. Vol. 100, 1). Academic Press, Boston MA et al. 1983, ISBN 0-12-393301-3 , 5.2.11.
^ Paul R. Halmos : Lectures on Boolean Algebra. Reprinted edition. Springer, New York NY et al. 1974, ISBN 0-387-90094-2 , § 21, Theorem 10.
^ Richard V. Kadison, John R. Ringrose: Fundamentals of the Theory of Operator Algebras. Special Topics. Volume 1: Elementary Theory (= Pure and Applied Mathematics. Vol. 100, 1). Academic Press, Boston MA et al. 1983, ISBN 0-12-393301-3 , 5.2.1.
^ Richard V. Kadison, John R. Ringrose: Fundamentals of the Theory of Operator Algebras. Special Topics. Volume 1: Elementary Theory (= Pure and Applied Mathematics. Vol. 100, 1). Academic Press, Boston MA et al. 1983, ISBN 0-12-393301-3 , 5.7.21.

[1] Richard V. Kadison , John R. Ringrose : Fundamentals of the Theory of Operator Algebras. Special Topics. Volume 1: Elementary Theory (= Pure and Applied Mathematics. Vol. 100, 1). Academic Press, Boston MA et al. 1983, ISBN 0-12-393301-3 , 5.2.10.

[2] Richard V. Kadison, John R. Ringrose: Fundamentals of the Theory of Operator Algebras. Special Topics. Volume 1: Elementary Theory (= Pure and Applied Mathematics. Vol. 100, 1). Academic Press, Boston MA et al. 1983, ISBN 0-12-393301-3 , 5.2.11.

[3] Paul R. Halmos : Lectures on Boolean Algebra. Reprinted edition. Springer, New York NY et al. 1974, ISBN 0-387-90094-2 , § 21, Theorem 10.

[4] Richard V. Kadison, John R. Ringrose: Fundamentals of the Theory of Operator Algebras. Special Topics. Volume 1: Elementary Theory (= Pure and Applied Mathematics. Vol. 100, 1). Academic Press, Boston MA et al. 1983, ISBN 0-12-393301-3 , 5.2.1.

[5] Richard V. Kadison, John R. Ringrose: Fundamentals of the Theory of Operator Algebras. Special Topics. Volume 1: Elementary Theory (= Pure and Applied Mathematics. Vol. 100, 1). Academic Press, Boston MA et al. 1983, ISBN 0-12-393301-3 , 5.7.21.