During my beginning PhD-research I have encountered the following question/problem:
- Let $X$ be a boolean space. Is it possible to reconstruct the space $X$ from its boolean algebra of clopen sets, $Cl(X)$ (viewed as a boolean algebra)?
By a boolean space I mean a (non-empty) zero-dimensional locally compact Hausdorff space, or equivalently, a Hausdorff space with a basis consisting of compact open sets. The particular spaces I am studying are second countable and have no isolated points as well -- in case it makes a difference.
The set of clopen subsets of $X$, $Cl(X)$, forms a boolean algebra under the usual set-operations of union, intersection and complement. It is generally not a complete boolean algebra.
In the case that $X$ is in fact compact (i.e. a Stone space) the answer is of course yes, this being the classical Stone duality. In the locally compact case however, it seems to me like the answer in general is no, based on the references below, as these seem to require the boolean algebra of compact open sets instead. And it is not possible to "detect" compactness of an element $A \in Cl(X)$, as one only has finite unions/joins in this boolean algebra.
- Dimov – Some Generalizations of the Stone Duality Theorem (http://rmi.tsu.ge/tolo2/presentations/Dimov.pdf)
- Doctor – The Categories of Boolean Lattices, Boolean Rings and Boolean Spaces (https://cms.math.ca/openaccess/cmb/v7/cmb1964v07.0245-0252.pdf)