# The Boundary of a Countable Union of Almost Disjoint Cubes

1. Sep 26, 2010

### 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 {Ri} 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.

2. Sep 26, 2010

### Office_Shredder

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

3. Sep 26, 2010

### Dr_Noface

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