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Math Help - some topology questions about closure and boundary

  1. #1
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    some topology questions about closure and boundary

    so i was reviewing topology and i came across some questions that i was not able to prove:

    1)show that for a set E in R^n, the boundary of E is contained in the closure of E.
    the given definition is that x is a boundary point of E if x is in the complement of int(E) U int(complement of E), and that the closure of E is the intersection of all closed sets containing E. i tried picking an x in the boundary of E, but couldnt relate it to the closure in anyway through set theoretic methods.

    2) show that for a set E in R^n, closure of E = E U (the boundary of E).
    again i know that the way to do it is set inclusion, but still have trouble relating the boundary of E to the closure of E.

    thanks!
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  2. #2
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    Quote Originally Posted by squarerootof2 View Post
    1)show that for a set E in R^n, the boundary of E is contained in the closure of E.
    the given definition is that x is a boundary point of E if x is in the complement of int(E) U int(complement of E), and that the closure of E is the intersection of all closed sets containing E. i tried picking an x in the boundary of E, but couldnt relate it to the closure in anyway through set theoretic methods.
    It is sufficient to show that if C is a closed set containing E then x\in C where x is a boundary point of E.

    If x\in (\text{int}(E) \cup \text{int}(E^*))^* then x\in ( \text{int}(E)^* \cap \text{int}(E^*)^*).
    Thus, x\in \text{int}(E)^* and x\in \text{int}(E^*)^*.

    Note E\subseteq C \implies C^* \subseteq E^* \implies \text{int}(C^*) \subseteq \text{int}(E^*).

    Since C is closed it means C* is open therefore \text{int}(C*) = C^*.

    Therefore, C^* \subseteq \text{int}(E^*) \implies \text{int}(E^*)^* \subseteq (C^*)^* = C

    It follows that x\in \text{int}(E^*)^* \subseteq C
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  3. #3
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    i was thinking about it and i think i found a better version of the proofs. can we just say boundary of E is contained in boundary of the closure of E, which should be contained in the closure of E as E is a closed set?
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