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Math Help - Path connectedness and connectedness

  1. #1
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    Path connectedness and connectedness

    Problem

    Show that an open set \Omega \in \field{C} is pathwise connected if and only if it is connected.

    Attempt at Solution

    Suppose \Omega is not connected and suppose \Omega is pathwise connected. Then, for any points a,b \in \Omega there exists a continuous function f which connects a and b together. Let V,W \subset \Omega, where V \cap W = \emptyset and \Omega = V \cup W. Let a \in V and b \in W. Clearly, we have a contradiction because there cannot exist a continuous function from a to b if \Omega is disconnected. Thus \Omega must be connected.

    How would I prove the other direction? Is this correct, btw?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Path connectedness and connectedness

    Quote Originally Posted by My Little Pony View Post
    Problem

    Show that an open set \Omega \in \field{C} is pathwise connected if and only if it is connected.

    Attempt at Solution

    Suppose \Omega is not connected and suppose \Omega is pathwise connected. Then, for any points a,b \in \Omega there exists a continuous function f which connects a and b together. Let V,W \subset \Omega, where V \cap W = \emptyset and \Omega = V \cup W. Let a \in V and b \in W. Clearly, we have a contradiction because there cannot exist a continuous function from a to b if \Omega is disconnected. Thus \Omega must be connected.

    How would I prove the other direction? Is this correct, btw?
    More generally, if V is a normed vector space then any connected open subset U of V is path connected.

    This follows from the fact that ever locally path connected and connected space is path connected. But, we know that U is connected and since every point of U has some open ball B containing it sitting inside U, but since B is convex and thus path connected you have that U is locally path connected.

    If you want to learn more, or see proofs of the above you can look at my blog post here (sorry, the formatting on that one is pretty bad).
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