Union of non empty sets

In summary, if you take the intersection of non-empty open subsets in Rn as n tends to infinity, the intersection can be empty or non-empty depending on the sets involved. In general, if the sets are compact and nested, the intersection will never be empty. However, there are cases where the intersection can be empty, such as when the sets are open intervals with decreasing lengths. The title of the thread may also be a bit misleading as the intersection is being discussed, not the union.
  • #1
Bachelier
376
0
If you take the intersection of non empty open subsets in Rn as n tends to infinity, such that

[tex]U_1 \supseteq U_2 \supseteq U_3... [/tex]

Is it empty?
 
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  • #2
not necessarily
 
  • #3
but it can. right?
 
  • #4
An example of that in R would be the sequence of sets:

[tex]A_n=\left(0,\frac{1}{n}\right)[/tex]

Then

[tex]\bigcap_{n=1}^{\infty}A_n=0[/tex]

In general if your sets are compact and nested, then the intersection will never be empty.
 
  • #5
Bachelier said:
but it can. right?

Sure it can happen. Here is a case where it does not. Let Ui be the closed interval of real numbers from zero to 1 + 1/n. The infinite intersection is the closed interval from zero to 1.
 
  • #6
Great. Thanks.
 
  • #7
The fact that this thread was titled "Union of non-empty sets" was a bit confusing!
 
  • #8
HallsofIvy said:
The fact that this thread was titled "Union of non-empty sets" was a bit confusing!


I posted the question right before I went to bed. LOL :tongue:
 

What is the definition of "Union of non empty sets"?

The union of non empty sets is a mathematical operation that combines all the elements from two or more sets into a single set without any duplicates. It is denoted by the symbol ∪.

How is the union of non empty sets different from the intersection of sets?

The union of non empty sets includes all the elements that are present in at least one of the sets, while the intersection of sets includes only the elements that are common to all the sets.

Is the union of non empty sets commutative and associative?

Yes, the union of non empty sets is commutative and associative. This means that the order in which the sets are combined does not affect the final result, and it can be grouped in any way without changing the outcome.

Can the union of non empty sets be empty?

Yes, the union of non empty sets can be empty if there are no common elements between the sets being combined. In this case, the resulting set will have no elements.

How is the union of non empty sets used in real life?

The union of non empty sets is used in various fields such as statistics, computer science, and economics. It is used to represent the combination of different data sets, the merging of multiple databases, and the calculation of probabilities in decision making.

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