The Boundary of a Countable Union of Almost Disjoint Cubes

[Dr_Noface]

Let E be a subset of R² that is non-empty, compact, and connected. Suppose furthermore that E is the union of a countably infinite number of almost disjoint closed cubes {R_i} with non-zero volume.

Is there anything interesting about this set, particularly its boundary? Can it have infinite length, for example? I can't think of very much to say about it, at all.

 

[Office_Shredder]

What do you mean by almost disjoint? Their intersection has measure zero?

 

[Dr_Noface]

My apologies, yes. If two cubes are almost disjoint then only their boundaries intersect.