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Math Help - Open sets and interior points Complex

  1. #1
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    Open sets and interior points Complex

    In complex analysis, how do I show that a set S is open iff each point in S is an interior point.


    Thank you much.
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  2. #2
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    The interior of a set S is the largest open set contained in S. If S is open clearly the interior of S is S so every point of S is in the interior of S since they are the same.

    Conversely if every point of S is an interior point then S \subset S^{\circ}. By definition of interior you always have the reverse containment. So since S=S^{\circ} S is open.
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  3. #3
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    Quote Originally Posted by nankor View Post
    In complex analysis, how do I show that a set S is open iff each point in S is an interior point.
    The approach to this question depends upon the definitions that your textbook uses.
    A text by Churchill is a fairly typical undergraduate complex analysis text.
    That text uses the statement of this problem as the definition of ‘open set’.

    Begin with the idea of \varepsilon-neighborhood: \varepsilon  > 0\; \Rightarrow \;N_\varepsilon  (z_0 ) = \left\{ {z:\left| {z_0  - z} \right| < \varepsilon } \right\}.

    In almost all developments, open sets as well as interior points are defined by \varepsilon-neighborhoods.
    So you need to follow the way that your textbook gives the definitions.
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