Is this Set Open? - Analysis of X and A in the Real Number System

[Spriteling]

Homework Statement

Take X = $$\Re$$ with the metric d(x,y) = |x-y|.

Let A = $$\bigcup^{\infty}_{n=1} \left( \frac{-1}{n},\frac{n+1}{n} \right)$$

Determine whether this set is open or closed.

Homework Equations

The intersection of a finite number of open subsets of X is open.

The Attempt at a Solution

Well, I am pretty sure that it's open, as it seems like the intersection goes to (0,1) as n goes to infinity. However, I'm unsure because it's an infinite intersection, and the rule only applies to finite intersections...

 

[Dick]

0 isn't in the open interval (0,1). Is it in the intersection?

 

[Spriteling]

Well, as n goes to infinity, -1/n goes to 0, and it's approaching 0 from below. So I would say that yes, it is in the intersection...meaning that the intersection is going to be [0,1] rather than (0,1), meaning that it's closed. Is that correct?

 

[Dick]

Well, as n goes to infinity, -1/n goes to 0, and it's approaching 0 from below. So I would say that yes, it is in the intersection...meaning that the intersection is going to be [0,1] rather than (0,1), meaning that it's closed. Is that correct?

That would be correct. The intersection of an infinite number of open sets is not necessarily open.

 

[Spriteling]

Yeah, I knew that it was only open for sure if it was a finite intersection.

Thank you for the hint!

 

[SammyS]

Homework Statement

Take X = $$\Re$$ with the metric d(x,y) = |x-y|.

Let A = $$\bigcup^{\infty}_{n=1} \left( \frac{-1}{n},\frac{n+1}{n} \right)$$

Determine whether this set is open or closed.

Homework Equations

The intersection of a finite number of open subsets of X is open.

The Attempt at a Solution

Well, I am pretty sure that it's open, as it seems like the intersection goes to (0,1) as n goes to infinity. However, I'm unsure because it's an infinite intersection, and the rule only applies to finite intersections...

What you have posted is an infinite union, not an infinite intersection.

$$\bigcup^{\infty}_{n=1} \left( \frac{-1}{n},\frac{n+1}{n} \right)=(-1,\,2)\cup(-1/2,\,3/2)\cup(-1/3,\,4/3)\cup\dots=(-1,\,2)$$

Did you mean to post the following instead?

$$\text{A}=\bigcap^{\infty}_{n=1} \left( \frac{-1}{n},\frac{n+1}{n} \right)$$

This infinite intersection is not the same as (0, 1). ‒1/n < 0 < (n+1)/n for all positive integers, n. So, 0 is in the infinite intersection. Similarly, 1 is also in this intersection.