The Boundary of a Countable Union of Almost Disjoint Cubes

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SUMMARY

The discussion focuses on the properties of the boundary of a set E in R2, which is a non-empty, compact, and connected union of a countably infinite number of almost disjoint closed cubes {Ri} with non-zero volume. The term "almost disjoint" is clarified to mean that the intersection of any two cubes has measure zero, implying that only their boundaries intersect. The participants explore whether the boundary of such a set can possess infinite length, indicating a complex geometric structure.

PREREQUISITES
  • Understanding of compact and connected sets in topology
  • Familiarity with measure theory and the concept of measure zero
  • Knowledge of geometric properties of closed cubes in R2
  • Basic concepts of boundaries in mathematical analysis
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  • Research the properties of compact sets in topology
  • Study measure theory, focusing on measure zero sets
  • Explore the geometric implications of almost disjoint sets
  • Investigate the concept of boundaries in higher-dimensional spaces
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Mathematicians, particularly those specializing in topology and measure theory, as well as students and researchers interested in geometric properties of sets in R2.

Dr_Noface
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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.
 
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What do you mean by almost disjoint? Their intersection has measure zero?
 
My apologies, yes. If two cubes are almost disjoint then only their boundaries intersect.
 

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