# Thread: fi nite intersection property

1. ## fi nite intersection property

Hey, I am really stuck on a question so if anyone can help me I would be very grateful.

Show that the set of finite unions of closed intervals $\{ \cup^n_{i=1} [x_i,y_i] :$ $0 \le x_i \le y_i \le 1$ $n \in \mathbb{N} \}$ in $X = [0,1]$ has the f.i.p.

I know that for a collection of sets to have the finite intersection property means that each nonempty finite subcollection of these sets has a nonempty intersection.

But I am unsure how to do this question and how to set it out.

Thanks in advance for any help.

2. Originally Posted by Nguyen
Hey, I am really stuck on a question so if anyone can help me I would be very grateful.

Show that the set of finite unions of closed intervals $\{ \cup^n_{i=1} [x_i,y_i] :$ $0 \le x_i \le y_i \le 1$ $n \in \mathbb{N} \}$ in $X = [0,1]$ has the f.i.p.

I know that for a collection of sets to have the finite intersection property means that each nonempty finite subcollection of these sets has a nonempty intersection.

But I am unsure how to do this question and how to set it out.

Thanks in advance for any help.

As you defined that set of intervals the claim is false: for example, it could be that two of the

intervals are $[1,\,1/3]\,,\,[1/2,\,2/3]$ , whose intersection is finite...

Tonio

3. Originally Posted by tonio
As you defined that set of intervals the claim is false: for example, it could be that two of the

intervals are $[1,\,1/3]\,,\,[1/2,\,2/3]$ , whose intersection is finite...

Tonio

Thanks for your reply but I still don't understand how to show that the set of finite unions has a f.i.p. Sorry for being a pain.

4. Originally Posted by Nguyen
Show that the set of finite unions of closed intervals $\{ \cup^n_{i=1} [x_i,y_i] :$ $0 \le x_i \le y_i \le 1$ $n \in \mathbb{N} \}$ in $X = [0,1]$ has the f.i.p.
Originally Posted by Nguyen
I still don't understand how to show that the set of finite unions has a f.i.p. Sorry for being a pain.
Of course you do not, Because as originally posted the statement is false.

5. Originally Posted by tonio
As you defined that set of intervals the claim is false: for example, it could be that two of the

intervals are $[1,\,1/3]\,,\,[1/2,\,2/3]$ , whose intersection is finite...
tonio meant "empty" here.

Tonio

6. Originally Posted by HallsofIvy
tonio meant "empty" here.
The intersection of [1/3, 1] and [1/2, 2/3] is [1/2, 2/3] isn't it?

-Dan