Topology #5 Open Sets
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Topology #4 Open Ballsby ThoughtSpaceZero3,971 views- 7:36
Topology #8 Unions and Intersections of Open Sets In Metric Spacesby ThoughtSpaceZero2,423 views
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Topology #10 Topology Examplesby ThoughtSpaceZero4,048 views
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Topology #9 Topological Spacesby ThoughtSpaceZero3,297 views
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Topology #2 Metric Examples (Part 1)by ThoughtSpaceZero4,045 views
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Topology #1 Metricsby ThoughtSpaceZero9,243 views
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Topology #15 Properties of Closed Setsby ThoughtSpaceZero1,440 views
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Topology #19 Neighbourhoods and Interiorsby ThoughtSpaceZero1,611 views
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Topology #14 Closed Setsby ThoughtSpaceZero2,309 views
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Topology #17 Subbasesby ThoughtSpaceZero1,145 views
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Topology #18 Topological Distinguishabilityby ThoughtSpaceZero1,033 views
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Topology #27 Netsby ThoughtSpaceZero710 views
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Topology #26 Directed Setsby ThoughtSpaceZero564 views
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Topology #29 Net Continuityby ThoughtSpaceZero929 views
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Topology #20 Neighbourhood Basesby ThoughtSpaceZero1,291 views
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Topology #28 Subnetsby ThoughtSpaceZero635 views
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General Topology Lecture 1 Part 2by vivictorov2,183 views
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Introduction to Topologyby jadenbane37,921 views
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