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Math Help - Need help understanding interior points, boundary points, isolated points, etc.

  1. #1
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    Wink Need help understanding interior points, boundary points, isolated points, etc.

    I am taking a Real Analysis class and we are learning about interior points, boundary points, isolated points and accumulation points.
    I really am not quite grasping what all of these are and the textbook I have isn't much help. I was wondering if anyone knows of a particular website that clearly explains each of these things or gives some examples of each type.
    Or, if you clearly understand them you could just type out what exactly each one is, explain how to decipher each one from another, and so on, to help me out.
    Thanks for any help!
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  2. #2
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    Interior points of a set A are points in A that have a little neighborhood around them that's also completely in A.

    Boundary points are points, not necessarily in A (though they could be), such that any neighborhood around a boundary point has at least one point in the set, and at least one point not in the set.

    Isolated points are points in A such that if you take a small enough neighborhood around them, the only points in that small enough neighborhood that are in A is the isolated point.

    Accumulation points are points, not necessarily in A (though they could be), such that, no matter how small a neighborhood around them you examine, there will always be other points of A in that neighborhood.

    So, you can see that interior points are always accumulation points, and are never boundary points or isolated points.

    Boundary points are never interior points, but they could be isolated, and they are always accumulation points.

    Isolated points are never interior points, but they are always boundary points. Isolated points are never accumulation points, because accumulation points must have other points in every neighborhood.

    Finally, accumulation points could be interior or boundary points. But they are never isolated.

    Does all that make sense?
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