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Math Help - Concept Clarification

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    is up to his old tricks again! Jhevon's Avatar
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    Concept Clarification

    I made this comment in one of my other posts, but because it wasn't the main problem, it was overlooked. I have deleted it from that post (to avoid double posting) and have, evidently, posted it here

    I want to clarify the concept of what it means to be "open."

    as i said, the definition of an isolated point (see here) bothers me. i was under the impression that a set consisting of a single point was always closed.

    an example given in my text (Set Theory and Metric Spaces by Kaplansky) says that the set X = \{ 1, \frac 12, \frac 13, \cdots , \frac 1n, \cdots \}, in its usual metric as a subset of the real line, is a discrete space with every point (and hence, every subset) is open.

    how do you reconcile this concept with the usual definition of being open?
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    Quote Originally Posted by Jhevon View Post
    (Set Theory and Metric Spaces by Kaplansky) says that the set X = \{ 1, \frac 12, \frac 13, \cdots , \frac 1n, \cdots \}, in its usual metric as a subset of the real line, is a discrete space with every point (and hence, every subset) is open. how do you reconcile this concept with the usual definition of being open?
    Are you confusing X = \{ 1, \frac 12, \frac 13, \cdots , \frac 1n, \cdots \} the subset of R^1 with X as a metric space by itself? As a subset of R^1, no subset of X is open. But as a space in and of itself every subset of X is open.

    To reconcile this idea think of the ball centered at 1/4 with radius 1/25. Is that a subset of {1/4}? Is that not the definition of an open set? So {1/4} is an open set in the metric space X.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Plato View Post
    To reconcile this idea think of the ball centered at 1/4 with radius 1/25. Is that a subset of {1/4}? Is that not the definition of an open set? So {1/4} is an open set in the metric space X.
    that's the thing. technically i would think it is not a subset of {1/4}. the ball would be outside the point. the only part of the ball that has anything in common with {1/4} is the center.

    i take it that we consider the ball to be a subset of {1/4} if its radius does not surpass half the distance between {1/4} and its closest neighbors? if so, why wouldn't the book mention a concept like that. it would seem that they expect it to be obvious to the reader, but it's not, at least not to me. i guess it was the concept of {1/4} as a subset of R^1 that was confusing me. metric spaces is something new to me after all, not too sure how to navigate through that world yet
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    Quote Originally Posted by Jhevon View Post
    the ball would be outside the point. the only part of the ball that has anything in common with {1/4} is the center.
    But that is exactly the point. In that discrete space 1/4 is the only point in the ball centered at 1/4 with radius 1/25. Every point is isolated. Each singleton set is both open and closed.
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