Χώρος Stone
- Ένας Μαθηματικός Χώρος.
Ετυμολογία[]
Η ονομασία "Χώρος Banach" σχετίζεται ετυμολογικά με το όνομα του μαθηματικού "Stone".
Εισαγωγή[]
A totally-disconnected compact space (X,T) whose field of all open-and-closed sets is isomorphic to Boolean algebra B.
This space is defined canonically from Boolean algebra B in the following way:
Boolean algebra B is the set of all ultrafilters of Boolean algebra B, while the topology T is generated by the family of subsets of the form:
- UA = {ξ∈X: A∈ξ},
- where A is an arbitrary element of Boolean algebra B.
Instead of ultrafilters, the set of maximal ideals, of two-valued homomorphisms or of two-valued measures on BB with an appropriate topology may be used.
Isomorphic Boolean algebras have homeomorphic Stone spaces.
Every totally-disconnected compact space is the Stone space of the Boolean algebra of its open-and-closed sets.
A form of Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to the Boolean algebra of clopen sets of a Stone space. This isomorphism forms a category-theoretic duality between the categories of Boolean algebras and Stone spaces.
The concept of a Stone space and its basic properties were discovered and studied by M.H. Stone (1934–1937).
- The Stone space of a Boolean algebra is metrizable if and only if the Boolean algebra is countable.
- A Boolean algebra is complete if and only if its Stone space is extremally disconnected (i.e. if the closure of any open set in the space is open).
- The perfect Cantor set is the Stone space of a countable atomless Boolean algebra (they are all isomorphic).
- The generalized Cantor discontinuum DmDm is the Stone space of the free Boolean algebra with mm generators.
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