# Why are free Boolean topological groups Hausdorff?

Assume $$X$$ is a Tychonoff space. Then $$A(X)$$ is the free topological abelian group over $$X$$. I know that $$A(X)$$ is Hausdorff and the canonical embedding from $$X$$ to $$A(X)$$ is a topological embedding.

Now consider the subgroup $$N:=2A(X)=\left\{g+g\ \colon\ g\in A(X)\right\}$$. The quotient $$B(X):=A(X)/N$$ is the free topological Boolean group. I want to show that $$B(X)$$ is Hausdorff which is equivalent to say that $$N$$ is closed in $$A(X)$$. Unfortunately, I do not know how show the closedness of $$N$$ since I can only describe the topology on $$A(X)$$ by the universal property and I do not know how that helps me here... Or is this maybe not true for a general Tychonoff space $$X$$ and one needs additional properties on $$X$$ like hereditarily disconnected?

I have found a reference on Free Boolean Groups

Genze, L.V. Free Boolean topological groups. Vestn. Tomsk. Gos. Univ. 2006, 290, 11–13

which maybe could answer my question but I am unable to find this text on math.sci.net (or anywhere else).

Thank you!

You can use the universality property with the following Boolean group as codomain: $$B$$ is the measure algebra over the unit interval (the quotient of the $$\sigma$$-algebra of Lebesgue measurable sets by the ideal of sets of measure zero), with symmetric difference as operation this is a Boolean group and $$d(A,B)=\lambda(A\Delta B)$$ defines a metric that will turn it into a topological group. This group contains a copy of the unit interval, namely the set $$\{[0,t]:0\le t\le 1\}$$. Since $$X$$ is Tychonoff you now have enough continuous functions to separate all non-trivial words from the empty word. This survey by Sipacheva gives more information: Free Boolean Topological Groups, Axioms 4 (2015) 492-517
• Thanks for this wonderful answer! I know the survey by Sipacheva but I could only find statements for spaces with ind$X$=0 there... Maybe I did not look close enough. Anyway, thank you for the $\sigma$-algebra example! Jul 26, 2021 at 9:08