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Thread: Point sets

  1. #1
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    Point sets

    Prove that, if S is open, each of its points is a point of accumulation of S.
    How can I prove this??
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathsohard View Post
    Prove that, if S is open, each of its points is a point of accumulation of S.
    How can I prove this??
    This is not true. Take any discrete space, then any singleton is open yet not a limit point.
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    This is not true??? hmm... I am using Advanced calculus by taylor and mann and on page 517 #5 they asked me to prove
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathsohard View Post
    This is not true??? hmm... I am using Advanced calculus by taylor and mann and on page 517 #5 they asked me to prove
    It's clearly untrue in arbitrary spaces, but since this is an advanced calculus book I'm going to take a wild guess and assume you meant and open subset of $\displaystyle \mathbb{R}^n$. If so then this is true if $\displaystyle S$ is non-empty. Let $\displaystyle s\in S$ and $\displaystyle U$ any neighborhood of it, then since $\displaystyle S$ is open we know there exists a neighborhood $\displaystyle N$ of $\displaystyle s$ such that $\displaystyle N\subseteq S$. Note then that $\displaystyle N\cap S$ is a neighborhood of $\displaystyle s$ contained in $\displaystyle s$. Choose some open ball $\displaystyle s\in B_{\delta}(s)\subseteq S\cap N$ and use the uncountability of open balls in euclidean space to find a point different from $\displaystyle s$ in $\displaystyle B_{\delta}(s)$.
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    Quote Originally Posted by mathsohard View Post
    This is not true??? hmm... I am using Advanced calculus by taylor and mann and on page 517 #5 they asked me to prove
    @mathsohard, you should give us the total context of a question.
    I just looked up that question. Not only is it not a general top-space it is a question about subsets of the real number line.
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