Basic Set Theory (Indexed Collection of Sets)

[Swixi]

Homework Statement

Give an example of an indexed collection of sets {A_{\alpha} : \alpha\in\Delta} such that each A_{\alpha}\subseteq(0,1) , and for all \alpha and \beta\in\Delta, A_{\alpha}\cap A_{\beta}\neq \emptyset but \bigcap_{\alpha\in\Delta}A_{\alpha} = \emptyset.

Homework Equations

None.

The Attempt at a Solution

I've found a solution that is:
Let A_{\alpha}=(0, \frac{1}{\alpha}), where \alpha\in\Delta=\mathbb{N}

and my main problem is that I don't understand how this is possible.

I understand that A_{\alpha}\cap A_{\beta}\neq \emptyset for any alpha/beta because the intersection will always be (0, 1/max(alpha,beta)).

But I don't understand how \bigcap_{\alpha\in\Delta}A_{\alpha} = \emptyset is true. Wouldn't every set in the family have the smallest real in it, because every set would be (0, a)? I feel like I'm probably not thinking about this the right way.

Thanks for any help!

 

[ZioX]

By definition, if x is in the intersection then x is in (0,1/n) for every n in N. Equivalently, 0<x<1/n. As n-> infty, 1/n -> 0. Using the N-epsilon definition of limits we know that if we let epsilon=x then there is a natural N such that 1/N<x. Therefore, x is not in A_N. Contradiction.

The intersection of any finite number of A_\alpha will be (0,min{1/n_1,1/n_2,...,1/n_k}) but if we want to to intersect all of the sets, {1,1/2,1/3,1/4,...} has no minimum.

Essentially we are probing at the properties of strictly decreasing chains of open sets.

 

[Swixi]

Thanks for the help.

When I was working on this, I initially figured that the intersection over all of the sets is empty like you proved (because n-> infinity and 1/n -> 0).

But, if you were to choose that set "at infinity" (which makes \bigcap empty) as your A_\alpha, then wouldn't its intersection with any A_\beta be empty too?

Can you explain to me why this is wrong? I understand how this works for any finite amount of sets, but my intuitive understanding of infinite sets feels quite weak.

 

[ZioX]

I don't understand your problem. Maybe you are thinking of this in terms of limits?

\mbox{lim}_{k\to\infty}\cap_{n=1}^k\left(0,\frac{1}{n}\right)=\mbox{lim}_{k\to\infty}\left(0,\frac{1}{k}\right)=(0,0)=\emptyset

Severe abuse of notation.

 

[Swixi]

I think it's just my grasp of infinite sets that is weak.

I suppose I need to do more work with families of sets.

Thanks again.