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Math Help - Limit point Definition

  1. #1
    Member mohammadfawaz's Avatar
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    Limit point Definition

    hey all,
    I read the following definition of a limit in some lectures notes:
    "Suppose that A⊆X (in a metric space). The point x_0 \in X is a limit point of A if for every neighborhood U(x_0, \epsilon) , of xo, the set U(x_0, \epsilon) is an infinite set."
    Well, the definition I know is: x_0 is a limit point of a set A if every deleted neighborhood of x_0 intersects A in at least one point.
    Can someone explain to me the first definition? I can't make the link between the two definitions
    thanx
    Last edited by mohammadfawaz; July 20th 2010 at 04:27 AM.
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  2. #2
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    If U(x_0;\epsilon) infinitely many points then then U(x_0;\epsilon)\setminus\{x_0\} contains at least one point.

    What don't you understand?
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  3. #3
    Member mohammadfawaz's Avatar
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    OK, but any neighborhood U(x_0;\epsilon)\setminus\{x_0\} contains at least one point anyway. It's a neighborhood. They don't mention intersection with A here! That's what I'm not getting.
    thanx
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  4. #4
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    Well I think the it should be understood that it is infinitely many points of A.
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