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Thread: 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 $\displaystyle B \subset X $ with x, y in $\displaystyle B$. 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 $\displaystyle B \subset X $ with x, y in $\displaystyle B$. 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 $\displaystyle \bar{A}$ 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 $\displaystyle C_x$ be the connected component of X containing x. For points x,y in X, the connected component $\displaystyle C_x$ containing x and the connected component $\displaystyle C_y$ containing y are either identical or disjoint. Thus, each point $\displaystyle x \in X$ belongs to exactly one connected component $\displaystyle C_x$ which is the largest connected subset of X containing x.

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

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