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Math Help - Topology Question - Regions

  1. #1
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    Topology Question - Regions

    Suppose that G is a region and let a \in G. Prove that G\{a} is a region.

    So far I have divided the proof up into 2 parts, showing that G\{a} is open and showing that it is connected. I have done the open part, but am stuck on the connectedness.
    I have assumed for contradiction that G\{a} is disconnected (generally seems to work for this type of question given the definition of connectedness), but cannot see where to get the contradiction from. Help much appreciated.
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  2. #2
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    Re: Topology Question - Regions

    Quote Originally Posted by Speed1991 View Post
    Suppose that G is a region and let a \in G. Prove that G\{a} is a region.
    How is region defined?
    For example in Moore Spaces regions are points sets.
    That is axiom 0.
    But in other spaces there are more conditions.
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    Re: Topology Question - Regions

    A region is defined as a non-empty connected subset of  \mathbb{C}
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    Re: Topology Question - Regions

    Quote Originally Posted by Speed1991 View Post
    A region is defined as a non-empty connected subset of  \mathbb{C}
    If G\subset \mathbb{C} is an open set, then G is connected if and only if any two points of G may be connected by a curve in G (in fact we can connect any two points of G by a chain \Gamma of horizontal and vertical segments lying in G) . Now, consider the cases a\not\in \Gamma and a\in\Gamma .
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