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Anoother two questions

  1. Feb 25, 2008 #1
    1. What's the difference between PERFECT SET and COMPLETE SET? Can I have an explicit explain to it, rather than a discribtion from definition?

    2. How can I verify whether a set is compact or not more evidently and effective?

    Tks!
     
    Last edited: Feb 25, 2008
  2. jcsd
  3. Feb 25, 2008 #2

    HallsofIvy

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    First, are you talking about topology? From your second question, I guess that you are but there are notions of 'perfect' and 'complete' sets for many different fields of mathematics. Topologically, a set is "perfect" if it is non-empty, closed, and has no isolated points. I don't recognize "complete set" in topology and cannot find any reference to it. Plenty of references to "complete set" of different kinds of things!
    Do you have a definition of "complete set"?

    Precisely what is your definition of "compact" set? One way to determine if a set is compact is by showing that every infinite sequence of points in the set has a subsequence that converges to a point in the set. If you are talking about subsets of Euclidean spaces, then, of course, a set is compact if and only if it is both closed and bounded.
     
  4. Feb 26, 2008 #3
    I mentioned "complete set", it was definded as: a set where every Cauchy sequence is convergent, from Chapter 2 of W. Rudin's Principle of Mathematical Analysis .

    And "compact set" is definded as: every open cover of the set has a finite subcover. Thus the compact set I was talking about is of general difinition. And thanks for reminding me of the Euclidean one.
     
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