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Math Help - Theorem on Closure

  1. #1
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    Theorem on Closure

    This is prob 7 on pg 56 of "Taylor, General Theory of Functions and Integration, First Ed, 1965"

    Prove 3). Hint: Prove 4) and observe that when this is combined with 2) we get 3).


    1) Definition of Closure. S' is the set of all accumulation points of S

    2) Definition of boundary of S

    3)

    4)

    S0 is interior of S

    COMMENT. I like the style, level, and brevity of the first few chapters. I just want a feel for fundametals of analysis, like Heine-Borel. But the author is constantly adding theorems to be proved by reader, without answers. This drives me nuts. I give the above as a sample problem. I don't even know where to start. All I can do is give words to the formulas and show they make intuitive sense.
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  2. #2
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    The essence of the problems you posted is understanding definitions.

    A point is interior to a set if some open set containing the point is a subset of the set. An intuitive way to look at it is: a point is interior if every point close to it is also interior. Note too the existential, some open set.

    On the other hand, a point is a boundary point if every open set containing the point also contains a point in the set and a point not in the set. That is: every open set about the point intersects both the set and its complement.

    Therefore, no interior point can be a boundary point and visa versa.
    Thus it is easy to apply definitions to prove
    \begin{gathered}<br />
 2)~~ \beta (S) = \overline S \backslash S^o  \hfill \\ \\<br />
  3)~~\beta (S) = \overline S  \cap \overline {S^c }  \hfill \\ <br />
\end{gathered}

    You may also learn some things by viewing the LaTeX.
    It really is best to learn to code.
    Last edited by Plato; December 14th 2010 at 01:11 PM.
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