Is the Intersection of an Infinite Collection of Open Sets Always Open?

[jinsing]

Homework Statement

a) If U and V are open sets, then the intersection of U and V (written U \cap V) is an open set.
b) Is this true for an infinite collection of open sets?

Homework Equations

Just knowledge about open sets.

The Attempt at a Solution

a) Let U and V be open sets, x be in U \cap V, and a,b be real numbers. Then there exists open intervals (a1,b1) in U and (a2,b2) in V such that x is in (a1,b1) and x is in (a2,b2). Now let (a,b) = (a1,b1) \cap (a2,b2). Since x is in (a1, b1) and x is in (a2,b2), then x is in (a,b) and by definition is an open interval. Moreover, since (a,b) is a subset of U \cap V, then U \cap V is an open set.

Is there anything I'm missing from this proof, like proving that the intersection (a1,b1) \cap (a2,b2) is an open interval? Or is that overdoing it..?

b) I know this isn't true (the intersection from n=1 to infinity of all open sets (-1/n, 1/n) = {0}, which is not an open set), but again I feel like I'm missing something - specifically proving that {0} is the intersection of all of those open sets, and proving {0} is not open.

I know these questions are incredibly straightforward, but my professor is kind of a stickler on the little details, especially the ones that seem pretty obvious. Thanks for the help!

 

[micromass]

a) Let U and V be open sets, x be in U \cap V, and a,b be real numbers. Then there exists open intervals (a1,b1) in U and (a2,b2) in V such that x is in (a1,b1) and x is in (a2,b2). Now let (a,b) = (a1,b1) \cap (a2,b2). Since x is in (a1, b1) and x is in (a2,b2), then x is in (a,b) and by definition is an open interval. Moreover, since (a,b) is a subset of U \cap V, then U \cap V is an open set.

This is good. However, it would be nice to state in the beginning that you are working with open sets in \mathbb{R}. Your argument does not hold for other spaces (as the open sets there are not necessarily generated by intervals).

Is there anything I'm missing from this proof, like proving that the intersection (a1,b1) \cap (a2,b2) is an open interval? Or is that overdoing it..?

This is crucial. I would include it.

b) I know this isn't true (the intersection from n=1 to infinity of all open sets (-1/n, 1/n) = {0}, which is not an open set), but again I feel like I'm missing something - specifically proving that {0} is the intersection of all of those open sets, and proving {0} is not open.

OK, you got the point. Now, where are you stuck? On proving that intersection, or proving that {0} isn't open??

By the way, it is possible to type LaTeX here. See this thread https://www.physicsforums.com/showthread.php?t=546968

 

[jinsing]

I'm basically stuck on proving the intersection (for both parts a and b) and proving 0 isn't open. I think I need just a shove in the right direction..they seem so self-explanatory to me that I don't know how to go about proving any of these things too rigorously.

 

[micromass]

OK, let's start by proving that {0} isn't open. We need definitions for this. How did you define open??

 

[jinsing]

A set S is open if for all x in S there is an open interval (a,b) contained in S with x in (a,b).

Could we say that for any ε > 0 there is no open interval (0-ε, 0+ε) that is contained within {0}, so {0} isn't open? Or something like that?

 

[micromass]

A set S is open if for all x in S there is an open interval (a,b) contained in S with x in (a,b).

Good. Let's apply this definition on {0}. Clearly, our x=0. So we must find an open interval (a,b)\subseteq \{0\} such that 0 is in (a,b). But then (a,b)=\{0\} has one element. Can you derive a contradiction from this?? (for example, by showing that (a,b) has more than one element).

For the intersection questions. You need to prove

(a,b)\cap (c,d)=(\max\{a,c\},\min\{b,d\})

and

\bigcap_{n\in \mathbb{N}_0}{(-1/n,1/n)}=\{0\}

These are just equalities of sets. Do you know how to show an equality of sets?? To prove A=B, just pick an x in A and show that it is in B and pick an x in B and show that it is in A.