# Stone topological Boolean algebras

I am looking for an initial reference for a theorem which is known, namely:

Theorem: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological algebra) iff $A$ is isomorphic to $\mathcal{P}(S)$, where $S$ is a set and a power-set $\mathcal{P}(S)$ is equiped with product topology.

The proposition can be found e.g. in book Stone spaces (Corollary in VI.3.2 Stone-type dualities pp. 247 of 2002 edition) by P. Johnstone. I am going to cite it in a paper but it is given there without any reference on the original, initial proof, in a fashion of a folklore result.

I heаrd somewhere that it is Strauss theorem (but don't know which Strauss is it, is it Dona Strauss, may be not). Anyway I need to cite an author and paper.

Thanks for kindness and effort in advance.

Edit: Dave L Renfro pointed me to the paper of Dona Strauss where the following result is proven by means of Pontryagin duality.:

Theorem: If a Boolean algebra $A$ admits a compact Hausdorff topology (i.e. is the underlying algebra of a compact Hausdorff topological algebra) then $A$ is isomorphic to $\mathcal{P}(S)$ where $S$ is a set and a power-set $\mathcal{P}(S)$ is equipped with product topology.

as well he pointed me to the paper of Guram Bezhanishvili and John Harding, where them provide simplified proof without use of Pontryagin duality. On that I remembered that I once even discussed the paper with John. Since the last result is enough for me, I accept the answer. Thank you Dave L Renfro.

• @YCor I'll edit post little bit. – Evgeny Kuznetsov Feb 23 '18 at 16:42
• Possibly the paper you want is: Dona Papert Strauss, Topological lattices, Proceedings of the London Mathematical Society (3) 18 #2 (April 1968), 217-230. The paper is behind a paywall and I do not have access to it (without travelling to a nearby university library). However, Strauss' paper is cited in On the proof that compact Hausdorff Boolean algebras are powersets by Guram Bezhanishvili and John Harding, which is freely available. – Dave L Renfro Feb 23 '18 at 19:02
• Note that every compact Hausdorff topology on the underlying additive group of $A$ is totally disconnected, so "$A$ admits a Stone space topology" is equivalent to "$A$ has a topology of topological algebra that is compact Hausdorff". – YCor Feb 23 '18 at 19:12
• Thanks for the question and answer! A few weeks ago I googled to see whether there's a notion of compact Boolean algebra with such a theorem and unfortunately this terminology "compact Boolean algebra" is used in another meaning. So I'm happy to have the answer now without even requesting it! – YCor Feb 23 '18 at 21:29
• Pontryagin duality is something very standard and even easier to prove in this particular context (compact/discrete elementary abelian 2-groups). Is the original proof really more complicated? – YCor Feb 23 '18 at 23:27