This is a curiosity question that I've been grappling with as I've been reading more about lattice theory:
Is it possible to endow some connected subset of $\mathbb{R}^n$ with a generalized boolean algebra structure?
Here you can choose the $n$ and any nontrivial subset. I'm looking for a subset of $\mathbb{R}^n$ where we can define meet and join operations, relative complements, etc in a way which is compatible with typical boolean algebra operations.
As this is pretty open-ended, here are a few things that I've considered so far:
It's well known that the set of divisors of a squarefree natural number forms a boolean algebra, with meet and join operations given by the GCD and LCM operations. This is close to what I want as it is a subset of $\mathbb{R}$, but the natural numbers aren't connected.
I've looked into Riesz spaces which also seemed promising. Example 1.5 in the aforementioned considered two vectors $x,y \in \mathbb{R}^n$ and defined an elementwise partial ordering, with the meet and join operations given by elementwise infimum and supremum. This yields a vector lattice, but it's not quite a generalized boolean algebra as I can't figure out any way of doing complements, even relative ones.
As it seems all finite dimensional Riesz spaces are isomorphic to the above (classification theorem), it seems to me that there's no hope of getting a proper boolean algebra on all of $\mathbb{R}^n$. But I'd also be happy with any nice subset of $\mathbb{R}^n$ that doesn't necessarily have to be closed under vector operations. For example, I've been toying with the sphere of unit-norm vectors, and have been attempting to define a complemented lattice structure where "1" is the north pole and "0" is the south pole, but to no avail.
I'm pretty stuck and would appreciate any further ideas! Perhaps there's some clever order or lesser-known space from the literature that I'm not aware of. Any references or tangentially related ideas are also very welcome.