# 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. Feb 23, 2018 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. Feb 23, 2018 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, 2018 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, 2018 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, 2018 at 23:27