1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Are limit points and interior points of a set contained in the set?

  1. Oct 16, 2011 #1
    1. The problem statement, all variables and given/known data

    Just wondering if I'm understanding the definitions correctly. I honestly feel like an idiot for asking this.

    2. Relevant equations

    A point p is an interior point of E is there is a neighborhood N of p such that N contained in E.

    A point p is a limit point if the set E if every neighborhood of p contains a point qp such that q is in element of E.

    3. The attempt at a solution

    It looks like an interior point of E must be contained in E, but a limit point of E is not necessarily contained in E. Am I right?
     
  2. jcsd
  3. Oct 16, 2011 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, you are.
     
  4. Oct 16, 2011 #3
    Sweet. Keep on standby for more questions.
     
  5. Oct 16, 2011 #4
    Just tell me if these seems legit. I sense something wrong about it.

    Let Eo denote the set of all interior points of a set E.
    (a) Prove that Eo is always open.
    (b) Prove that E is open if and only if Eo = E
    (c) If G is a subset of E and G is open, prove that G is a subset of Eo


    screen-capture-3-30.png
     
  6. Oct 16, 2011 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Well, your first proof is not doing well. You say Nr(p) is not contained in E0. Just because E0 is contained in E that doesn't mean Nr(p) isn't contained in E. E is larger than E0. That's probably what's bothering you. The whole proof is needlessly indirect. Just use that E0 is a union of open sets.
     
    Last edited: Oct 16, 2011
  7. Oct 17, 2011 #6
    Is this an okay start, brah? Where should I got from here? I'm trying to prove that every point in Eo is an interior point by choosing an arbitrary element of Eo and showing that it cannot not be an interior point.

    screen-capture-4-15.png
     
  8. Oct 17, 2011 #7

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Read it. It's false. If p is not an interior point of E0 then every neighborhood contains at least one point in E0^C. It's certainly not necessarily true that it's CONTAINED in E0^C. Can't you think of a way to write E0 as a union of open sets?
     
  9. Oct 17, 2011 #8
    Sorry, brah. Let me give this another go.

    If E is open, then clearly Eo is open.
    If E is closed, then there exists an element p such that p is in E and (Eo)c.
    Suppose Eo is closed.
    -----> (Eo)c is open
    -----> There is an r >0 s.t. Nr(p) is contained in (Eo)c


    ...... So close! I'm trying to derive a contradiction to the fact that Eo is all interior points of E (my only premise to contradict).
     
  10. Oct 17, 2011 #9

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    This isn't going well. And I don't think you are very close. If p is in E0 then there is an open ball Nr(p) that is contained in E, yes? Can you think of a reason why every other point p' in Nr(p) is also contained in E0?? Tell me that first. I don't think the 'contradiction' proof is working for you.
     
  11. Oct 18, 2011 #10
    Just thinking about it visually, if I choose any point p' in Eo, then the neighborhood Nr-d(p,p')(p) will surely contain only points of E.
     
  12. Oct 18, 2011 #11

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Sure, that works. So every point in E0 has a neighborhood contained in E0.
     
  13. Oct 18, 2011 #12
    How do I show that? Triangle Inequality? I somehow need to show that a point q is in Eo if d(q,p') < r - d(p,p').
     
  14. Oct 18, 2011 #13

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    It's not really necessary to do that. And you probably don't want to assume you are in a metric space. If N is an OPEN neighborhood of p what do you know about every point p' in N?
     
  15. Oct 18, 2011 #14
    There is an open neighborhood centered at p' that is contained in the open neighborhood centered at p.
     
  16. Oct 18, 2011 #15

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Ok, so can you use that to prove E0 is open?
     
  17. Oct 18, 2011 #16
    I'm pretty sure we can assume E is a subset of a metric space, because all the definitions we're supposed to use (neighborhood, interior point, limit point, etc.) relate to metric spaces.
     
  18. Oct 18, 2011 #17
    Yeah. I've shown that an arbitrary point in N is an interior point of N. Thus N is open and Eo, the union of all interior points, is open.
     
  19. Oct 18, 2011 #18

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    You needn't use metric spaces. Proofs work for general topological spaces. The definitions you gave in post #1 are probably picked up from Kelley's book. Newer books define the interior as an open set.
     
  20. Oct 18, 2011 #19

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    That's not very well stated at all.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Are limit points and interior points of a set contained in the set?
Loading...