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Betti number

  1. Jan 31, 2009 #1
    This is probably a silly question but I am not from a maths background. I am a bit confused about betti numbers. From what i know :
    The zeroth betti number is the number of connected components in space
    The first betti number is the number of loops in space
    the second is the number of cavities in the space

    However does this mean that the zeroth betti number is always atleast one since there is always one connected component ?
     
  2. jcsd
  3. Jan 31, 2009 #2
    Why would there be always at least one connected component ? A topological space can have more than one components. One can always call the collection of two distinct spheres as one space.
     
  4. Jan 31, 2009 #3

    Hurkyl

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    And two is at least one. :tongue:
     
  5. Jan 31, 2009 #4
    Can you give me a case where the zeroth Betti number is 0 and explain why please?
     
  6. Jan 31, 2009 #5
    Oops. I misunderstood the statement. :rofl: My bad.
     
  7. Feb 1, 2009 #6
    The empty space has no connected components. :)
     
  8. Feb 4, 2009 #7
    the empty set is connected
     
  9. Feb 4, 2009 #8
    There is always one connected component.

    But your interpretation of the first Betti numer as the number of loops is hasty. What is the first Betti number of the projective plane? What about the Klein Bottle? How many loops do they have?
     
  10. Feb 4, 2009 #9
    Yes, it is connected, but it has no connected components. A connected component must be nonempty.
     
  11. Feb 4, 2009 #10
    true, i just thought i'd clarify the terminology
     
  12. Feb 18, 2009 #11
    Totally agree :)
     
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