Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Nowhere dense subset of a metric space

  1. Feb 18, 2010 #1
    Can we have some examples in which a nowhere dense subset of a metric space is not closed?
     
  2. jcsd
  3. Feb 18, 2010 #2
    just take a Cauchy sequence without its limit point e.g. 1/2^n
     
  4. Feb 25, 2010 #3
    Is zero not a limit point of 1/2^n since as n gets large, 1/2^n goes to zero?
     
  5. Feb 25, 2010 #4

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    "Is zero not a limit point of 1/2^n since as n gets large, 1/2^n goes to zero? "

    Yes, and that is precisely the issue here. A closed subset of a metric space

    (I think this is true in any topological space)contains all its limit points. One

    way of seeing this is seeing what would happen if the limit point L of a closed

    set C in X was not contained in C. Then L is in X-C, and every 'hood (neighborhood)

    of L in X-C , intersects points of C.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Nowhere dense subset of a metric space
  1. Metric Spaces (Replies: 3)

Loading...