The Boundary of a Countable Union of Almost Disjoint Cubes

In summary, the boundary of a countable union of almost disjoint cubes is the set of points that are shared by two or more cubes in the union, and it is important in defining the shape and structure of the union and has a measure of zero in relation to the concept of measure. It can be calculated by finding the shared points on the edges and faces of the cubes, and it is possible for it to be empty if there are no shared points between the cubes.
  • #1
Dr_Noface
3
0
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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  • #2
What do you mean by almost disjoint? Their intersection has measure zero?
 
  • #3
My apologies, yes. If two cubes are almost disjoint then only their boundaries intersect.
 

FAQ: The Boundary of a Countable Union of Almost Disjoint Cubes

1. What is the boundary of a countable union of almost disjoint cubes?

The boundary of a countable union of almost disjoint cubes is the set of points that lie on the edges and faces of the cubes. It is the collection of points that are shared by two or more cubes in the union.

2. Why is the boundary of a countable union of almost disjoint cubes important?

The boundary of a countable union of almost disjoint cubes is important because it helps to define the shape and structure of the union. It also plays a significant role in various mathematical and scientific applications, such as in topology and geometry.

3. How is the boundary of a countable union of almost disjoint cubes calculated?

The boundary of a countable union of almost disjoint cubes can be calculated by finding the points that are common to two or more cubes in the union. This can be done by examining the edges and faces of the cubes and identifying the shared points.

4. Can the boundary of a countable union of almost disjoint cubes be empty?

Yes, it is possible for the boundary of a countable union of almost disjoint cubes to be empty. This occurs when there are no shared points between any of the cubes in the union, meaning that the cubes are completely disjoint from each other.

5. How does the boundary of a countable union of almost disjoint cubes relate to the concept of measure?

The boundary of a countable union of almost disjoint cubes is related to the concept of measure in that it has a measure of zero. This means that it has no volume or area, and therefore does not contribute to the overall measure of the union. However, it still plays a crucial role in defining the shape and structure of the union.

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