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Math Help - 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 \mathbb{R}^n. If so then this is true if S is non-empty. Let s\in S and U any neighborhood of it, then since S is open we know there exists a neighborhood N of s such that N\subseteq S. Note then that N\cap S is a neighborhood of s contained in s. Choose some open ball 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 s in 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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