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!

A set is closed iff it equals an intersection of closed sets

  1. Sep 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Let M be a metric space, A a subset of M, x a point in M.

    Define the metric of x to A by

    d(x,A) = inf d(x,y), y in A

    For [itex]\epsilon[/itex]>0, define the sets

    D(A,[itex]\epsilon[/itex]) = {x in M : d(x,A)<[itex]\epsilon[/itex]}

    N(A,[itex]\epsilon[/itex]) = {x in M: d(x,A)[itex]\leq[/itex][itex]\epsilon[/itex]}

    Show that A is closed iff A = [itex]\bigcap[/itex]N(A,[itex]\epsilon[/itex]) for [itex]\epsilon[/itex]>0

    2. Relevant equations



    3. The attempt at a solution

    I was able to do the <= implication.
    I'm having trouble doing the => one...

    If I assume A is closed, all I know is that its compliment is open, which is a union of open sets... then should I just use that fact and DeMorgans laws to show A is in fact an intersection of those closed N(A,[itex]\epsilon[/itex])?

    OR should I take an arbitrary point in A, and show it belongs to the intersection..??

    Just need some help on what method I should try to use.

    Thanks!
     
  2. jcsd
  3. Sep 28, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    So, for [itex]\Rightarrow[/itex], you must prove that

    [tex]A=\bigcap{N(A,\varepsilon)}[/tex]

    There are 2 things to prove now: [itex]\subseteq[/itex] and [itex]\supseteq[/itex]. Only [itex]\supseteq[/itex] is nontrivial. For this: take a point in the intersection, and prove that it is in A.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: A set is closed iff it equals an intersection of closed sets
  1. Closed set (Replies: 1)

  2. Closed set (Replies: 3)

  3. Closed sets (Replies: 8)

  4. Closed set (Replies: 11)

  5. Closed set (Replies: 14)

Loading...