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Math Help - limit points of metric space

  1. #1
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    limit points of metric space

    In my text book, the definition of a limit point is as follows:
    If X is a metric space, a point x of X is a limit point of the subset A of X if every epsilon-neigborhood of x intersects A in at least one point different from x. An equivalent condition is to require that every neigborhood of x contain infinitely many points of A.

    Does that mean that, for example, all the points in the interior of an open circle are limit points of that circle?

    In a more specific example,
    let A={ 0<(x^2)+(y^2)<=1} where A is a subset of R^2
    let Lim(A) be the set of all limit points of A (I am assumingthis is what Lim(A) means btw)
    is Lim(A)={ 0<=(x^2)+(y^2)<=1} where Lim(A) is a subset of R^2?

    but then the closure of A, Cl(A)=Lim(A) which makes me question my understanding of the definition of limit points...or is that right?

    Intuitively, I think of Lim(A)={0} union { x^2+ y^2=1} which is actually Bd(A)
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  2. #2
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    Quote Originally Posted by 234578 View Post
    In my text book, the definition of a limit point is as follows:
    If X is a metric space, a point x of X is a limit point of the subset A of X if every epsilon-neigborhood of x intersects A in at least one point different from x. An equivalent condition is to require that every neigborhood of x contain infinitely many points of A.

    Does that mean that, for example, all the points in the interior of an open circle are limit points of that circle?
    Yes. All the points of the interior of the circle are limit points in the circle. However, not all the limit points of the circle are in the interior of the circle.

    Specifically, in \mathbb{R}^2, the closed circle \{x \in \mathbb{R}^2 : d(x,x_0) \le r\} where x_0 is the center and r is the radius contains all of its limit points.


    In a more specific example,
    let A={ 0<(x^2)+(y^2)<=1} where A is a subset of R^2
    let Lim(A) be the set of all limit points of A (I am assumingthis is what Lim(A) means btw)
    is Lim(A)={ 0<=(x^2)+(y^2)<=1} where Lim(A) is a subset of R^2?

    but then the closure of A, Cl(A)=Lim(A) which makes me question my understanding of the definition of limit points...or is that right?

    Intuitively, I think of Lim(A)={0} union { x^2+ y^2=1} which is actually Bd(A)
    Your notation is hard to understand. I've never seem Lim(A). It may mean the set of all limit points?

    The closure of A is different from the set of all its limit points because it may contain its isolated points as well. In other words, the closure of A is the union of A and its limit points.
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  3. #3
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    For example, if A= \{(x, y)| x^2+ y^2< 1\}\cup\{(2, 0)\} the set of all limit points is the closed circle \{(x,y)| x^2+ y^2\le 1\}- all points interior to the circle (which were in A) and points on the circle (which were not in A) but does NOT include the "isolated point" (2, 0). The closure of A is A together with any limit points that were not in A: \{(x, y)| x^2+ y^2\le 1\}\cup \{(2, 0)\}.
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