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Math Help - Connected subset of a Topology Question

  1. #1
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    Connected subset of a Topology Question

    Let A be a subset of a topological space of X. Show that if C is a connected subset of X that intersects both A and X-A, then C intersects \mathfrak{d}(A)
    Last edited by mr fantastic; May 22nd 2009 at 04:08 AM. Reason: Restored deleted question
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  2. #2
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    Quote Originally Posted by flaming View Post
    Let A be a subset of a topological space of X. Show that if C is a connected subset of X that intersects both A and X -A, then C intersects \mathfrak{d}(A)
    Let int (A) be an interior of a set A.

    Since X = int (A) \cup \ \mathfrak{d}(A) \cup int(X \setminus A) and C intersects both A and X \ A,

    C=(C \cap int (A)) \cup (C \cap \mathfrak{d}(A)) \cup (C \cap int(X \setminus A)),

    Now suppose to the contrary that C \cap \mathfrak{d}(A) is empty. If C \cap \mathfrak{d}(A) is empty, then both C \cap int (A) and C \cap int(X \setminus A) are not empty (C should intersect both A and X\A). We know that int (A) and int  (X \setminus A) are not connected. It follows that C=(C \cap int (A)) \cup (C \cap int(X \setminus A)) is not connected, contradicting that C is a connected subset of a topological space X.

    Therefore, C intersects \mathfrak{d}(A).
    Last edited by aliceinwonderland; April 9th 2009 at 02:05 AM. Reason: Correction
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