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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


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    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


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    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.)
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