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Math Help - Show that connceted components are closed

  1. #1
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    Show that connceted components are closed

    Let (X, d) be a metric space and define a binary relation on X via

    x ~ y iff there is a connected set B \subset X with x, y in B. The equivalence classes are known as the connected components of X.

    Show that every connected component is closed.

    I tried to show that the complement of every connected component is open, but didn't get far.
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  2. #2
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    Quote Originally Posted by h2osprey View Post
    Let (X, d) be a metric space and define a binary relation on X via

    x ~ y iff there is a connected set B \subset X with x, y in B. The equivalence classes are known as the connected components of X.

    Show that every connected component is closed.

    I tried to show that the complement of every connected component is open, but didn't get far.
    Lemma 1. If a subset A of a metric space X is connected, then \bar{A} is connected.

    Since the connected components of a metric space X are equivalent classes in X, X can be partitioned into the connected components of X. Let C_x be the connected component of X containing x. For points x,y in X, the connected component C_x containing x and the connected component C_y containing y are either identical or disjoint. Thus, each point x \in X belongs to exactly one connected component C_x which is the largest connected subset of X containing x.

    Assume that C_x is a connected proper subset of \overline{C_x}. By lemma 1, \overline{C_x} is a connected subset containing x, contradicting that C_x is the largest connected subset of X containing x. Thus, C_x = \overline{C_x}.

    We conclude that \forall x \in X, C_x is closed.
    Last edited by aliceinwonderland; May 13th 2009 at 08:54 AM. Reason: Typo in lemma 1, etc.
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