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!

Metric Space, open and closed sets

  1. Nov 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Let X be set donoted by the discrete metrics
    d(x; y) =(0 if x = y;
    1 if x not equal y:
    (a) Show that any sub set Y of X is open in X
    (b) Show that any sub set Y of X is closed in y

    2. Relevant equations

    In a topological space, a set is closed if and only if it coincides with its closure. Equivalently, a set is closed if and only if it contains all of its limit points.

    A subset U of a metric space (M, d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x, y) < ε, y also belongs to U. Equivalently, U is open if every point in U has a neighbourhood contained in U.

    3. The attempt at a solution

    I tried to use the line y=x and said that there is a finite many balls centered along the line, that means its open because of the union. But my prof said that we r working in abstract so we cant use R2. How would this work then?
     
  2. jcsd
  3. Nov 19, 2009 #2
    For (b), it is clear that if we fix a point x, the set {y | d(x,y) < 1/2} is just the point x What are the limit points of X?

    For (a), use the fact that a set is open if it's complement is closed. If all the subsets of X are closed, then the complement of every subset is closed.
     
  4. Nov 19, 2009 #3
    (b) Show that any sub set Y of X is closed in X.....is the correct question
     
  5. Nov 19, 2009 #4
    Yes, I assumed that's what was meant.
     
  6. Nov 19, 2009 #5
    the set {y | d(x,y) < 1/2}.....how was developed? and why less than 1/2?
     
  7. Nov 19, 2009 #6

    lanedance

    User Avatar
    Homework Helper

    there are only really two cases here as the distance between every point is 1, the two cases are

    d>=1
    d<1
    consisder what points aere within a ball of each radius in each case

    the choice of d<1/2 was probably an abritray d<1 case to consider
     
  8. Nov 19, 2009 #7

    lanedance

    User Avatar
    Homework Helper

    this is known as the discrete topology of X
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Metric Space, open and closed sets
Loading...