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Thread: "Closed" set in a metric space

  1. #1
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    Unhappy "Closed" set in a metric space

    Suppose that (x_n) is a sequence in a metric space X such that lim x_n = a exists. Prove that {x_n: n E N} U {a} is a closed subset of X.

    =================================

    Let B(r,a)={x E X: d(x,a) < r} denote the open ball of radius r about a.
    Definition: Let D be a subset of X. By definition, D is open iff for all a E D, there exists r>0 such that B(r,a) is contained in D.
    Definition: Let F be a subset of X. F is called closed iff whenever (x_n) is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limit points of sequences in F)
    Theorem: F is closed in X iff the complement of F is open.

    I know the definitions, but I just don't know out how to construct the proofs formally. I am not sure how to prove that some set is "closed"...I can't find any similar examples in my textbook.

    May someone kindly help me out?
    Any help is appreciated!



    [note: also under discussion in math links forum]
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Accidental post, see next post.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kingwinner View Post
    Suppose that (x_n) is a sequence in a metric space X such that lim x_n = a exists. Prove that {x_n: n E N} U {a} is a closed subset of X.

    =================================

    Let B(r,a)={x E X: d(x,a) < r} denote the open ball of radius r about a.
    Definition: Let D be a subset of X. By definition, D is open iff for all a E D, there exists r>0 such that B(r,a) is contained in D.
    Definition: Let F be a subset of X. F is called closed iff whenever (x_n) is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limit points of sequences in F)
    Theorem: F is closed in X iff the complement of F is open.

    I know the definitions, but I just don't know out how to construct the proofs formally. I am not sure how to prove that some set is "closed"...I can't find any similar examples in my textbook.

    May someone kindly help me out?
    Any help is appreciated!



    [note: also under discussion in math links forum]
    Let $\displaystyle E$ be the set in question. Consider $\displaystyle E'$ and let $\displaystyle e\in E'$. Let $\displaystyle \delta=\frac{d(a,e)}{2}>0$ and consider $\displaystyle B_{\delta}(e),B_{\delta}(a)$ these are disjoint open sets about $\displaystyle e,a$ respectively. Since $\displaystyle x_n\to a$ we have that $\displaystyle B_{\delta}(a)$ contains all but finitely many points of $\displaystyle \{x_n\}$ say $\displaystyle x_1,\cdots,x_m$. Let $\displaystyle B_{\delta_k}(e)$ be the open ball of radius $\displaystyle \frac{d(e,x_k)}{2}$ and consider $\displaystyle O=B_{\delta}(e)\cap\bigcap_{k=1}^{m}B_{\delta_m}(e )$, clearly $\displaystyle O$ is an open set containing $\displaystyle e$ disjoint form $\displaystyle E$. The conclusion follows.
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  4. #4
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    In a metric space $\displaystyle (X,d)$ you should know that
    $\displaystyle D(A;x)=\inf\{d(a,x):a\in A\}$ is the distance the point $\displaystyle x$ is from the set $\displaystyle A$.
    Moreover, you should know that A closure $\displaystyle \overline{A}=\{x(A;x)=0\}$
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