(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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- Feb 28th 2009, 10:25 AMhorowitzSubspace 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. - Feb 28th 2009, 11:10 AMGammaDefinition
A) take a point in K, find a neighborhood about it which does not intersect anything else in K. let $\displaystyle \frac{1}{n}\in K$, consider $\displaystyle d = 1/2 [ 1/n - 1/(n+1)]$ and let $\displaystyle U=(1/n-d, 1/n +d)$

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.