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Math Help - Subspace topology

  1. #1
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    Subspace topology

    (a)Let K={1/n ∈ R| n∈Z+} Show that the standard topology on K is the discrete topology.
    (b) Let K*=K union {0}. Show that the standard topology on K* is not the discrete topology.
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  2. #2
    Super Member Gamma's Avatar
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    Iowa City, IA
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    Definition

    A) take a point in K, find a neighborhood about it which does not intersect anything else in K. let \frac{1}{n}\in K, consider d = 1/2 [ 1/n - 1/(n+1)] and let U=(1/n-d, 1/n +d)

    B) Try to find an open neighborhood about 0 that doesn't intersect K... Good luck, as the limit as n goes to infinity of the harmonic (1/n) is 0, so it is a limit point.
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