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Homework Help: Please check my Contrapositive statement

  1. Feb 14, 2013 #1
    I am trying to write a CP for:

    Every connected M. Space with at least 2 points is uncountable.


    if a MS X is connected with |X|≥ 2 => X is uncountable.


    a MS X has only one point => X is not connected.

    Last edited: Feb 14, 2013
  2. jcsd
  3. Feb 14, 2013 #2
    I guess the correct statement is that:
    if X has only one point it is separated.
  4. Feb 14, 2013 #3
    How do I prove this though. the part about the singleton
  5. Feb 14, 2013 #4


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    If a space has only one point, it's clearly countable isn't it? Forming the CP isn't the challenge, it's showing a metric space with two points in it that's pathwise connected is uncountable. Any ideas on that one?
  6. Feb 15, 2013 #5
    You seemed to have changed "connected" into 'pathwise" connected. The OP was talking about just connected. The statement is true in both cases though (as I'm 100% sure you know), but it's much easier to prove if you take pathwise connected.
  7. Feb 15, 2013 #6


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    Yep, true. I actually don't think it's hard for either form of connected. Thanks!
  8. Feb 15, 2013 #7


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    A metric space consisting of a single point is necessarily connected, is it not? The only non-empty open subset is the whole space.
  9. Feb 15, 2013 #8

    D H

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    Given the statement "if a then b" (equivalently, ab), the contrapositive is "if not b then not a" (¬b → ¬a).

    Your right hand side in the first statement is "X is uncountable". The negation is "X is countable", which is how your contrapositive statement should start.
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