# Re: Stone duality

```Hi Bruno,

I just found this statement that captures the basic idea at the root
of how Prattian Dualism "works":```
```

http://en.wikipedia.org/wiki/Stone_space

Stone spaces
Each Boolean algebra <http://en.wikipedia.org/wiki/Boolean_algebra>  B has
an associated topological space, denoted here S(B), called its Stone space.
The points in S(B) are the ultrafilters
<http://en.wikipedia.org/wiki/Ultrafilter>  on B, or equivalently the
homomorphisms from B to the two-element Boolean algebra
<http://en.wikipedia.org/wiki/Two-element_Boolean_algebra> . The topology on
S(B) is generated by a basis <http://en.wikipedia.org/wiki/Basis_(topology)>
consisting of all sets of the form

where b is an element of B.
For any Boolean algebra B, S(B) is a compact
<http://en.wikipedia.org/wiki/Compact_space>  totally disconnected
<http://en.wikipedia.org/wiki/Totally_disconnected>  Hausdorff
<http://en.wikipedia.org/wiki/Hausdorff_space>  space; such spaces are
called Stone spaces (also profinite spaces). Conversely, given any
topological space X, the collection of subsets of X that are clopen
<http://en.wikipedia.org/wiki/Clopen_set>  (both closed and open) is a
Boolean algebra.
Representation theorem
A simple version of Stone's representation theorem states that any Boolean
algebra B is isomorphic to the algebra of clopen subsets of its Stone space
S(B). The full statement of the theorem uses the language of category theory
<http://en.wikipedia.org/wiki/Category_theory> ; it states that there is a
duality <http://en.wikipedia.org/wiki/Duality_of_categories>  between the
category <http://en.wikipedia.org/wiki/Category_theory>  of Boolean algebras
<http://en.wikipedia.org/wiki/Boolean_algebra_(structure)>  and the category
of Stone spaces. This duality means that in addition to the isomorphisms
between Boolean algebras and their Stone spaces, each homomorphism from a
Boolean algebra A to a Boolean algebra B corresponds in a natural way to a
continuous function from S(B) to S(A). In other words, there is a
contravariant functor <http://en.wikipedia.org/wiki/Contravariant_functor>
that gives an equivalence
<http://en.wikipedia.org/wiki/Equivalence_(category_theory)>  between the
categories. This was the first example of a nontrivial duality of
categories.
The theorem is a special case of Stone duality
<http://en.wikipedia.org/wiki/Stone_duality> , a more general framework for
dualities between topological spaces
<http://en.wikipedia.org/wiki/Topological_space>  and partially ordered sets
<http://en.wikipedia.org/wiki/Partially_ordered_set> .
The proof requires either the axiom of choice
<http://en.wikipedia.org/wiki/Axiom_of_choice>  or a weakened form of it.
Specifically, the theorem is equivalent to the Boolean prime ideal theorem
<http://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem> , a weakened
choice principle which states that every Boolean algebra has a prime ideal.
***
Please note the direction of the mappings between a pair of Boolean algebras
and a pair of Stone spaces! They flow in opposite directions. This is the
essence of the duality between Time and Logic.

A -> B iff S(A) <= S(B)

When we sum over all of the possible arrows, the notion of direction itself
vanishes, thus one could misunderstand this as the vanishing of Time but it
is not, it is only that the possibility of measuring change has vanished.
Change in-it-self as eternal Becoming exists nonetheless.

Onward!

Stephen P. King

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