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Stupid question?

  1. Apr 17, 2007 #1
    1. The problem statement, all variables and given/known data
    Can a complete metric space have empty interior?


    2. Relevant equations
    In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M.


    3. The attempt at a solution
    But if M has no Cauchy sequence to start with or anything else for that matter (i.e have empty interior than it can also be labeled as complete? Or is my understanding lacking some important information?
     
  2. jcsd
  3. Apr 17, 2007 #2

    Dick

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    A single point constitutes a trivial metric space. It's complete, open, closed, compact and lots of other things, too!
     
  4. Apr 17, 2007 #3
    A single point also has empty interior.

    What about a metric space with 2 points? It still has empty interior.
     
  5. Apr 17, 2007 #4

    Dick

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    True. Guess I'd better think again.
     
  6. Apr 17, 2007 #5

    Dick

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    We'd better be a little careful here. Interior and exterior only have nontrivial meaning if we are speaking of the metric space as a subset of another space. If we are speaking of a single point space {x} in isolation then the interior of {x} is {x}. If we are speaking for example of {0} as a subset of the reals, then it has empty interior.
     
  7. Apr 17, 2007 #6
    Interior of the whole metric space is always non empty.

    So the subspace of a complete metric space is compelete so has non empty interior? Since we could have a sequence of points starting in the large metric space and obtaining a limit in this subspace. Where this limit point can be in the interior of the subspace. Hence non empty interior for this subspace?
     
  8. Apr 17, 2007 #7

    AKG

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    The interior of a metric space X is X itself. So a metric space has empty interior iff that metric space is itself empty. The empty set together with the empty function is a metric space.
     
  9. Apr 17, 2007 #8
    Is the empty set also complete?
     
  10. Apr 17, 2007 #9

    Dick

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    pivoxa15, this is a sad moment. Think!
     
  11. Apr 18, 2007 #10
    From the definition
    In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M.

    I'd say yes because there is no Cauchy sequence or any sequence in the empty set.
     
  12. Apr 18, 2007 #11

    Dick

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    I would agree. But are these void case problems really that interesting? Is the empty set colorless?
     
  13. Apr 21, 2007 #12
    That`s the right one
     
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