1. Not finding help here? Sign up for a free 30min 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!

Topology question; derived pts and closure

  1. Oct 26, 2008 #1
    1. The problem statement, all variables and given/known data
    If A is a discrete subset of the reals,

    prove that

    [tex]A'=cl_x A \backslash A[/tex]

    is a closed set.


    2. Relevant equations
    A' = the derived set of A
    x is a derived pt of A if [tex]U \cap (A \backslash \{x\}) \neq \emptyset[/tex] for every open U such that x is in U.

    Thrm1. A is closed iff A=cl(A)
    Thrm2. cl(emptyset)=emptyset


    3. The attempt at a solution

    "Proof". Using Thrm1 with A' we can see that A' is closed iff A'=cl(A'). Since A is a discrete subset of the Reals we know that the set consists of isolated pts. Since there are no derived pts in A, then A'= emptyset. Using Thrm2 we know that cl(emptyset)=emptyset. Therefore A'=cl(A'). And thus A' is closed.

    Does this proof work?
     
  2. jcsd
  3. Oct 26, 2008 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I think so. Is there any difference between cl_x(A) and cl(A)? A' is closed because it's empty.
     
  4. Oct 27, 2008 #3
    Well, a classmate pointed out to me that the set B={1/n : n in Naturals}, has a derived point of {0}. I feel like this set is discrete, yet it's derived points are not the empty set.

    I thought I had this one. Any ideas of where I have gone wrong? My guess would be assuming that discrete subsets have no derived points but then what direction do I take?
     
  5. Oct 27, 2008 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Whoa. You're right. You caught me. I thought the proof sounded a little vacuous. So you want to prove that if x is not in A', then there is a neighborhood of x that does not intersect A'. If x is in A, use that A is discrete. If x is not in A, then use that A is closed. Can you take it from there?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Topology question; derived pts and closure
  1. Closure question (Replies: 2)

Loading...