not appropriate to say B_r(x) = {x}. A more appropriate notation would be to say that {x} is a subset, because if they are equal then the ball is also a subset of {x}. (i.e. two sets are equal if and only if they are subsets of one another). Further, its inaccurate to say that the points are open sets on their own, in fact they are closed sets; you cant "fit" any open balls into a single point, so for that set (namely the single point), not "all" the points can balls that can fit in the set
@brydust The ball is most certainly a subset of {x}, as if it were to contain any other point other than x it would necessarily contain all points of the space, by definition of the metric. In the discrete topology all subsets are both open and closed.
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christinadidi10 3 days ago
not appropriate to say B_r(x) = {x}. A more appropriate notation would be to say that {x} is a subset, because if they are equal then the ball is also a subset of {x}. (i.e. two sets are equal if and only if they are subsets of one another). Further, its inaccurate to say that the points are open sets on their own, in fact they are closed sets; you cant "fit" any open balls into a single point, so for that set (namely the single point), not "all" the points can balls that can fit in the set
brydust 1 year ago
@brydust The ball is most certainly a subset of {x}, as if it were to contain any other point other than x it would necessarily contain all points of the space, by definition of the metric. In the discrete topology all subsets are both open and closed.
ThoughtSpaceZero 1 year ago 2