- #1

Swixi

- 3

- 0

## Homework Statement

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

## Homework Equations

None.

## The Attempt at a Solution

I've found a solution that is:

Let [tex]A_{\alpha}=(0, \frac{1}{\alpha})[/tex], where [tex]\alpha\in\Delta=\mathbb{N}[/tex]

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

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

But I don't understand how [tex]\bigcap_{\alpha\in\Delta}A_{\alpha} = \emptyset[/tex] 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!