Reviewing for Analysis Exam- can't figure out pf. about Set of Lim. Pts. being closed

In summary, the conversation discusses a theorem about limit points in a metric space and the speaker's struggle to find a rigorous proof for an upcoming exam. They consider different approaches and receive helpful hints. Ultimately, the speaker is able to solve the problem and thanks the other person for their advice.
  • #1

Homework Statement

Theorem: Given a metric space [itex]\left(X,d\right)[/itex], the set of all limit points of a subset [itex]E\subset X[/itex], denoted [itex]E'[/itex] is a closed set.

I have an Analysis Exam tomorrow and have been studying for quite awhile and last week, my professor gave us a list of Theorems to know the proofs of. This is one of them. I perused my notes and could not find the proof. I also searched through three or four Analysis texts and could not find a good proof. It doesn't seem like it would be hard to prove, but I can't think of a really rigorous one.

I have the following ideas:

- Show that [itex](E')^c[/itex] is open.
- I let there be a [itex]p \in (E')^c[/itex] and a [itex]q\in E'[/itex]
- From there I thought that possibly considering the two cases of [itex]q \in E[/itex] and [itex]q \notin E[/itex].
- From there I am stuck. I think I might need to use something about an open neighborhood and use distances between points, but I am not sure.

I really appreciate any helpful clues or hints. I am just looking to be pushed in a good direction, not for the solution to be given out. I would ask my professor, but I am not on campus today and he doesn't have office hours today.

Thanks Much
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  • #2

Well, if [itex]p \in (E')^c [/itex], then it is not a limit point. Thus, there exists a neighborhood around [itex]p[/itex] that does NOT contain any points of [itex]E[/itex]. Your job is to make the radius of this neighborhood sufficiently small so that this occurs. How does this imply that the set is open?

Another option is to explicitly prove that [itex]E'[/itex] is closed. This may be easier...
  • #3

Thank you for your response. I will go from there and see what I can come up with. I'll try both routes suggested.
  • #4

I had the Exam today, and a variation of this problem appeared. The professor wanted us to do the proof for sub sequential limits instead of general limit points. And I figured it out last night after lineintegral's advice (I took the second option of direct proof- turned out to be painfully obvious). So thanks much!

1. What is a "Set of Lim. Pts." in the context of reviewing for an analysis exam?

A "Set of Lim. Pts." refers to a set of limit points, also known as accumulation points or cluster points. In analysis, a limit point is a point that can be arbitrarily close to a given set of points, even if it is not a member of the set itself. This concept is important in understanding the properties of sets and functions in mathematics.

2. How does a set of limit points relate to closed sets?

A set of limit points is closed if it contains all of its limit points. In other words, every limit point of a closed set must also be a member of that set. This is an important property because it helps us determine if a set is closed or not, which has implications for continuity and convergence in analysis.

3. Is every closed set also a set of limit points?

No, not necessarily. A set can be closed without containing all of its limit points. However, every set of limit points is closed. This is because if a set contains all of its limit points, then it cannot have any limit points outside of itself, making it closed.

4. How can I determine if a set of limit points is closed?

To determine if a set of limit points is closed, you can use the definition of a closed set: a set is closed if it contains all of its limit points. This means that for every point in the set of limit points, there must be a sequence of points in the original set that converges to that point. If this condition is met, then the set of limit points is closed.

5. Why is understanding closed sets and sets of limit points important in analysis?

Understanding closed sets and sets of limit points is important in analysis because it helps us define and study the properties of continuous functions, sequences, and convergence. It also allows us to prove important theorems in analysis, such as the Bolzano-Weierstrass theorem and the intermediate value theorem. Additionally, it is a fundamental concept in topology, which is the study of continuity and convergence in mathematics.

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