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!

Closed, bounded, compact

  1. Mar 9, 2006 #1
    Could someone explain me how these three concepts hang together?

    (When is a set bounded but not closed, closed but not bounded, closed but not compact and so one?)
     
  2. jcsd
  3. Mar 9, 2006 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Examples (real line with usual topology).

    Bounded not closed: 0<x<1
    Closed not bounded or compact: 0<=x<oo.
     
  4. Mar 9, 2006 #3

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    in R^n, compact is equivalent to closed and bounded, so a closed set is bounded iff compact, and a bounded set is closed iff compact.

    in a metric space, compact is equivalent to complete and totally bounded.

    in R^n which is itself complete, closed is equivalent to complete, and since every bounded set in R^n has comoact closure, bounded is equivalent to totally bounded.

    a totally bolunded set is one in which everys equence ahs a cauchy subsequence, and a completes et one in which everyu cauchy sequence converges.

    the connection is that a compact metric space is one in which every sequence has a convergent subsequence. (i think. it has been a long time since i taught this course.)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Closed, bounded, compact
  1. Upper bound/Lower Bound (Replies: 10)

  2. Chernoff Bounds (Replies: 2)

  3. A compactness problem~ (Replies: 1)

  4. Boundness of sets (Replies: 7)

Loading...