I’m sorry for the title. Please don’t hurt me. I have a condition.
Stone duality is a result that says (in a certain sense) that Boolean algebras are the same as zero-dimensional compact Hausdorff spaces. (It’s not an equivalence of categories as I had previously claimed, as the resulting functor is contravariant, but it is an equivalence of categories if you replace one of them with their dual).
I’ll assume you know a little bit of topology (enough to decipher ‘compact Hausdorff’ anyway), but I’ll give a quick refresher on Boolean algebras.
One way to define them is purely ring theoretic. A Boolean algebra is a ring with 1 such that for every x we have x^2 = x.
These have various nice properties. In particular every element of a Boolean algebra is its own additive inverse, as 4x = 4x^2 = (2x)^2 = 2x, so 2x = 0. Also, multiplication is commutative as x + y = (x + y)^2 = x^2 + xy + yx + y^2 = x + xy + yx + y. So xy + yx = 0 and thus by the previous observation xy = yx.
Two important notes: Firstly, the only Boolean algebra which is a field is {0, 1} with the obvious operations. Secondly, the homomorphic images of Boolean algebras are Boolean algebras. So if B is a Boolean algebra and I is a maximal ideal then B/I is a field, and so {0, 1}. Thus for any x in B we have either x in I or 1 + x is in I.
The classic example of a Boolean algebra is P(X) for some set X, with the symmetric difference as + and intersection as multiplication. 0 is the empty set and 1 is X. In the finite case, every Boolean algebra is isomorphic to one of these. In the infinite case, something more interesting happens.
But first, a note. We can turn any Boolean algebra into a lattice (a partially ordered set in which every finite set has a sup and inf, or equivalently a certain type of algebraic structure) of a very special kind:
Define x <= y if xy = x. Then x^2 = x, so x <= x, if x <= y and y <= x then xy = x and yx = y then x = y because the algebra is commutative. If x <= y and y <= z then xz = (xy)z = x(yz) = xy = x. So, it's a partial order. Note that 1 is the largest element of the order and 0 is the smallest.
I’m going to claim the rest of the details to check without proof:
sup{x, y} = x + y + xy
inf{x, y} = xy
We write sup{x, y} = x / y and inf{x, y} = x / y
These two operations are associative, commutative and idempotent. Further they distribute over eachother.
Finally we have that for every x there is a unique y such that x / y = 1 and x / y = 0. (Specifically y = 1 + x). We write this x^c. Further, De Morgan’s laws hold in that (x / y)^c = x^c / y^c and (x / y)^c = x^c / y^c. Further x <= y iff y^c <= x^c.
The first part is just saying that this forms a lattice. The second part is saying that this is a distributive lattice, and the third is that it is complemented.
In the power set case we have that / is union and <= is subset containment as one might expect.
Given any complemeted distributive lattice we can get a Boolean algebra structure back out of it, by taking x + y = (x / y^c) / (y / x^c), the symmetric difference.
So, Boolean algebras are the same thing as complemented distributive lattices. This is nice, but it’s not the main point here.
It does however give us one useful way of recharacterising things.
Theorem: I <= B is an ideal of B iff it contains 0, is closed under / and whenever y in I and x <= y we have x in I.
I’ll leave this as an exercise, but it’s not hard. But applying the complement operator to it gives us a dual definition:
F <= B is a filter iff it contains 1, is closed under / and whenever y in I and y <= x we have x in I.
Filters turn out to be a very useful equivalent way of looking at ideals.
Boolean algebras come up an awful lot in their own right. A lot of natural things in logic and set theory can be interpreted as Boolean algebras, and there are rich classes of examples of them. Most often they come up as lattices rather than algebras, but because these two are the same thing we can cheerfully switch between them as we wish.
The major example (which will lead us to Stone duality) is the following:
Let X be a topological space and let A be the set of clopen subsets of X. Then A is closed under union, intersection and complement and so forms a Boolean subalgebra of P(X). We call this the clopen algebra of X. In general this may not tell us an awful lot about X, but in the case where X is zero-dimensional (there is a basis of clopen sets) it will.
Stone duality is going to say the following:
1) For every boolean algebra B there is a topological space which we will call the Stone space of B such that B is isomorphic to its clopen algebra.
2) Given a zero dimensional compact hausdorff space X, X is homeomorphic to the Stone space of its clopen algebra.
So for every zero dimensional compact Hausdorff space there is a unique Boolean algebra, and vice versa. Further, both constructions are (contravariant) functorial.
I’m not going to prove this, but I will explain how it works and why it gives a zero dimensional compact Hausdorff space.
The Stone space of B, S(B) is the set of all homomorphisms from B into {0, 1}, given the subspace topology from {0, 1}^B. Equivalently it’s the Zariski topology on the set of maximal ideals of B. This is compact because it’s a closed subspace, and zero-dimensional and hausdorff because {0, 1}^B is, and both properties are preserved under restriction. The fact that it’s functorial is obvious.
Sketch proof of why X is homeomorphic to the Stone space of its clopen set. Let x in X. Then { U in clop : x in U } is an ultrafilter on clop(X), so gives rise to a maximal ideal and so a homomorphism into {0, 1}. You can check this is a homeomorphism.
The fact that it’s a contravariant functor actually makes it more interesting, not less.
Here’s why:
The stone space of P(N) is the set of all maximal ideals of P(N). By complementing we may equivalently regard it as the set of all maximal filters on P(N). i.e. the set of all ultrafilters, which form the Stone-Cech compactification of N, beta N. It’s an easy check to see that this gives the right topology as well. The Stone Cech compactification is a very major area of study in topology, and the Stone duality gives some very powerful tools for studying it.
Of more interest is the remainder, beta N N. Now, under Stone duality this is not a subalgebra of P(N) but a quotient of it. Specifically it is the quotient P(N) / Fin, where Fin is the ideal of finite subsets of P(N).
This Boolean algebra has some nice properties. Firstly, it is atomless. An atom in a Boolean algebra is an element a such that x <= a implies that x = a or x = 0. P(N) has lots of atoms (all the singletons), but we've quotiented out by the ideal generated by the atoms (this doesn't always succeed in getting rid of atoms, as it can introduce new ones, but in this case it does). A Boolean algebra being atomless is equivalent to its Stone space having no isolated points.
Most importantly, it has something called the strong countable seperation property. If A, B are countable subsets of P(N)/Fin such that for any finite F, G with F <= A, G <= B we have sup F < inf G then there is an interpolating element x. i.e. one such that for a in A, b in B we have a < x < b. This is incredibly useful, because it lets us prove the following:
Let A, B be Boolean algebras such that B has the strong countable seperation property. Let C <= A a countable subalgebra. Let f : C -> B be an embedding, a in A and define C’ to be the algebra generated by C and a. Then we can extend f to an embedding f’ : C’ -> B.
(The proofs of both of these are a bit technical and a pain to write up in ascii, so I don’t intend to include them here. The proofs for all of this will get written up properly at some later date).
Now, this lets us prove Cool Things.
Theorem:
Let A, B be Boolean algebras such that |A| = aleph_1 and B has the strong countable seperation property. Then A embeds into B.
Proof:
You can probably do this via Zorn, but I find transfinite induction more natural here.
Enumerate A as {a_t : t < aleph_1}. We'll define a sequence of subalgebras A_t and embeddings f_t : A_t -> B such that if t < s then A_t <= A_s and f_s restricted to A_t is f_t, and Union A_t = A. Then the patching of these functions to A will give an embedding of A into B.
But doing this is easy. We take A_t to be the algebra generated by Union_{s < t} A_s union {a_t}. We have an embedding from the union of the previous subalgebras, so the extension lemma gives us an embedding of A_t. We pick one such embedding and call it f_t, continuing the induction.
QED
In particular this works with B = P(N)/Fin
This probably doesn’t sound especially exciting at first glance. But lets plug it into Stone duality and see what we get:
Let X be a zero dimensional compact Hausdorff space of weight <= aleph_1 (the weight of a topological space is the smallest cardinality of a basis), then X is a continuous image of beta N N.
Ok, this is still interesting. But zero dimensional spaces are a bit special – there are certainly interesting examples of them, but a lot of the spaces which we want to study aren’t zero dimensional. Fortunately, there’s a useful theorem of point set topology we can apply.
Every compact Hausdorff space of weight k embeds as a subspace of [0, 1]^k, as a consequence of Urysohn’s lemma. Recall that [0, 1] is a continuous image of {0, 1}^aleph_0. Hence [0, 1]^k is a continuous image of {0, 1}^k. Thus any compact Hausdorff space of weight k is a continuous image of a closed subspace of {0, 1}^k, and so the continuous image of a zero dimensional compact hausdorff space of weight k.
Ok, now it’s interesting. We’ve suddenly got a huge wealth of continuous maps from beta N N. This is Paracivenko’s first theorem:
Every compact Hausdorff space of weight <= aleph_1 is the continuous image of beta N N.
It gets better. We can use almost exactly the same proof to give something which will (combined with the continuum hypothesis) give a very powerful result:
Theorem: Let A, B be Boolean algebras with the strong countable seperation property and |A| = |B| = aleph_1. Then A and B are isomorphic.
Proof:
The place where we use the continuum hypothesis is in the very statement of the theorem. |P(N)/Fin| = 2^aleph_0 = aleph_1 by CH.
We need to adapt the previous argument. We’ll use a trick called a back and forth argument.
We’ll define increasing sequences A_t, B_t countable subalgebras of A and B. with Union A_t = A, Union B_t = B.
Further there will be embeddings as follows:
f_t : A_t -> B_t
g_t : B_t -> A_{t+}
Such that g_t f_t = id and f_{t+} g_t = id.
Then these will paste together to give maps f : A -> B, g : B -> A which are homomorphisms and mutual inverses, proving the result.
Now that we’ve got this setup, it’s really just a case of applying the countable extension lemma.
Suppose we’ve done this for for s < t. Let C be the union of g_s(B_s) over s < t. Then this is a countable Boolean subalgebra, and we have an embedding of it h : C -> B given by the inverses of the g_s being patched together. By the construction we have Union_{s < t} A_s <= C, and h|_{A_s} = f_s. Now, extend this map h to the algebra generated by C and a_t. Call this algebra A_t and the extension f_t : A_t -> B.
Now, we have that the image of f_t contains B_s for s < t. Define B_t to be the algebra generated by the image of f_t and b_t and g_t : B_t -> A to be an extension of the inverse of f_t to this algebra.
The induction continues.
Phew. I think I mangled that explanation slightly. Hopefully the idea that I’m trying to convey is reasonably clear.
Anyway, the point is that under the assumption of the continuum hypothesis this becomes “P(N)/Fin is the unique Boolean algebra of cardinality aleph_1 with the strong countable seperation property.”
This is nice, because it’s a simple short characterisation of something important. Having these will very often give you interesting or useful results.
Now, hit it with Stone duality…
The question is, what does the statement ‘has the strong countable seperation property’ become under Stone duality?
It turns to be something slightly awkward:
Definition: X is a Paracivenko space if it is a compact zero dimensional Hausdorff space with no isolated points, such that every pair of disjoint F_sigma sets have disjoint closures and every non-empty G_delta set has empty interior.
Theorem: B has the strong countable seperation property iff its Stone space is Paracivenko.
I’m not going to prove this.
So, our uniqueness result becomes: beta N N is the unique Paracivenko space of weight aleph_1.
This is actually extremely useful, because there are a wealth of Paracivenko spaces of weight aleph_1 (under CH), and this theorem tells us they’re all homeomorphic – so by studying one, we study all of them.
For example, (beta N N)^2 is Paracivenko, so is homeomorphic to beta N.
Seems trivial? It’s not. You can’t prove this result under ZFC.
This gives rise to some very rich theory in modern set theoretic topology – the study of beta N under the continuum hypothesis is extremely nice, and this leads on to a lot of interesting topological and combinatorial results…
…which I know almost nothing about. So I’m going to stop here.