Is the Intersection of Infinite Non-Empty Open Subsets Empty?

[Bachelier]

If you take the intersection of non empty open subsets in Rⁿ as n tends to infinity, such that

$$U_1 \supseteq U_2 \supseteq U_3...$$

Is it empty?

 

[lavinia]

not necessarily

 

[Bachelier]

but it can. right?

 

[sutupidmath]

An example of that in R would be the sequence of sets:

$$A_n=\left(0,\frac{1}{n}\right)$$

Then

$$\bigcap_{n=1}^{\infty}A_n=0$$

In general if your sets are compact and nested, then the intersection will never be empty.

 

[lavinia]

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.

 

[Bachelier]

Great. Thanks.

 

[HallsofIvy]

The fact that this thread was titled "Union of non-empty sets" was a bit confusing!

 

[Bachelier]

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