Is K Open or Closed? A Topology GRE Question

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Discussion Overview

The discussion revolves around a topology question related to the nature of a set K, defined as the union of subsets Tk, which are contained within disjoint open sets Ek. Participants explore whether K is open or closed, considering the implications of compactness and the properties of the Ek sets.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants suggest that K might be closed, referencing the compactness of the sets Tk and the bounded nature of their elements.
  • Others argue that while K is formed from compact sets, the union of an infinite number of closed sets is not necessarily closed, raising questions about the specific structure of the sets involved.
  • A participant notes that the Ek sets are disjoint and not open, which may influence the nature of K.
  • Another participant emphasizes the importance of considering limit points and their relation to the compactness of Tk.
  • Some participants caution against the binary classification of sets as open or closed, highlighting the existence of sets that are neither.
  • There is a humorous exchange about the potential for test questions to be misleading or ambiguous regarding the classification of sets.

Areas of Agreement / Disagreement

Participants express differing views on whether K is open or closed, with no consensus reached. The discussion includes multiple competing perspectives and acknowledges the complexity of the question.

Contextual Notes

Participants mention that the nature of K depends on the properties of the sets Tk and Ek, and the discussion reflects uncertainty regarding the implications of compactness and the definitions of open and closed sets.

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

I say closed, but I'm not sure.
 
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ralphhumacho said:
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:
Ek = B(0, k) - B(0, k-1), where B(0,k) is an open ball around the origin with diameter k. Now suppose that Tk is a subset of Ek for all k, where T is compact. Now let K be the union (1 to infinity) of all Tk. 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.
 
Notice that the closest a point in Tk can be to zero is k-1, and the farthest is k. But since Tk is compact, the distance from the farthest point in Tk 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 Tk and hence x is in Tk and thus K itself.

Also, note that the Ek's are not open, but they are disjoint
 
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.
 
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.
 
your comment is correct in general, except that i think i have already settled this particular matter.
 
jostpuur said:
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?"
 
If a question on a test asks whether a set is open or closed, it sure as hell better be open or closed
 
Office_Shredder said:
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 :wink:

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? :wink: :-p
 
  • #10
Office_Shredder said:
If a question on a test asks whether a set is open or closed, it sure as hell better be open or closed

HallsofIvy said:
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!?
 
  • #11
Tac-Tics said:
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
 
  • #12
Office_Shredder said:
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-(
 
  • #13
nevermind. Misread.
 
Last edited:

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