Is K Open or Closed? A Topology GRE Question

[ralphhumacho]

I'm not very good at topology but am reviewing it for the GRE Subject Test. Here's a question that I think I know, but would like to check with you guys.

We define:
E_k = B(0, k) - B(0, k-1), where B(0,k) is an open ball around the origin with diameter k. Now suppose that T_k is a subset of E_k for all k, where T is compact. Now let K be the union (1 to infinity) of all T_k. Is K open or closed?

I say closed, but I'm not sure.

 

[Tac-Tics]

I'm not very good at topology but am reviewing it for the GRE Subject Test. Here's a question that I think I know, but would like to check with you guys.

We define:
E_k = B(0, k) - B(0, k-1), where B(0,k) is an open ball around the origin with diameter k. Now suppose that T_k is a subset of E_k for all k, where T is compact. Now let K be the union (1 to infinity) of all T_k. Is K open or closed?

I say closed, but I'm not sure.

It sure sounds closed. In R^n, compact means closed and bounded. With each T_k being completely contained within disjoint open sets E^k, each T_k is disjoint. While the union of an infinite number of closed sets isn't in general closed, it seems like in this situation, it is, because the sets are all separated in some sense. It has something to do with compactness, but I'm not sure how.

If you have a cover of each T_k called C_k, then the union of all C_k covers K. But K clearly isn't compact itself, as it is unbounded.

Oh, when you say K is the union from k=1 to k=infinity, I'm assuming you mean all integral values of k, though I'm not sure if it even affects the problem.

 

[Office_Shredder]

Notice that the closest a point in T_k can be to zero is k-1, and the farthest is k. But since T_k is compact, the distance from the farthest point in T_k to zero is bounded away from k. Hence if you have a limit point of K, say x, then |x| is going to be in between some k-1 and k, and from there it's clear x is a limit point of T_k and hence x is in T_k and thus K itself.

Also, note that the E_k's are not open, but they are disjoint

 

[mathwonk]

think about the meaning of closed. take a sequence of elements in the set that converges and ask whether the limit lies in the set?

isn't it obvious that the limit must lie in a bounded set in the space? hence the sequence also lies in a bounded set, hence in a finite union of compact sets. that dooos it.

 

[jostpuur]

There exists sets which are not open and not closed, so it is always dangerous to start with question like "closed or open?". It seems that in this case the carelessness isn't leading into trouble, though.

 

[mathwonk]

your comment is correct in general, except that i think i have already settled this particular matter.

 

[HallsofIvy]

There exists sets which are not open and not closed, so it is always dangerous to start with question like "closed or open?". It seems that in this case the carelessness isn't leading into trouble, though.

Indeed there also exist sets that are both open and closed so, strictly speaking one should always ask "open, closed, neither, or both?"

 

[Office_Shredder]

If a question on a test asks whether a set is open or closed, it sure as hell better be open or closed

 

[jostpuur]

If a question on a test asks whether a set is open or closed, it sure as hell better be open or closed

You never know

Why couldn't a professor write a question "Which one of A and B is correct?", and then give zero points for both A and B answers, the actual correct answer being something else?

 

[Tac-Tics]

If a question on a test asks whether a set is open or closed, it sure as hell better be open or closed

Indeed there also exist sets that are both open and closed so, strictly speaking one should always ask "open, closed, neither, or both?"

What if it's neither!?

 

[Office_Shredder]

What if it's neither!?

That's not a question choice. If the question writer gives you information, you have to assume it's true. When you see a question like: "Given an element in the field of rational numbers..." do you prove that the rational numbers are a field? When you're asked to calculate an integral of a function, do you prove that the function is integrable? Did you even bother to prove that it was a function you're integrating? What a waste of time that would be

 

[Tac-Tics]

That's not a question choice. If the question writer gives you information, you have to assume it's true. When you see a question like: "Given an element in the field of rational numbers..." do you prove that the rational numbers are a field? When you're asked to calculate an integral of a function, do you prove that the function is integrable? Did you even bother to prove that it was a function you're integrating? What a waste of time that would be

Whoops. I misread HallsofIvy's post. I thought he had left out "neither" in the list of possibilities, so I added it X-(

 

[redrzewski]

nevermind. Misread.