Let (X,d) be a metric space. prove that: (X,d) is discrete if only if X∩X′=∅,X′ is the set of all limit points of X
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HINT: $X\cap A=A$ for every subset $A$ of $X$, so what is $X'$?
Brian M. Scott
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in discrete, $X' $ is emty? ? – ILoveMath Nov 02 '13 at 06:40
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1@DonAnselmo: Yes, because ${x}$ is open for each $x\in X$. – Brian M. Scott Nov 02 '13 at 06:41
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how about the other direction? – ILoveMath Nov 02 '13 at 06:42
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@DonAnselmo: Let $x\in X$, and let $A=X\setminus{x}$. If $X'=\varnothing$, can $x$ be a limit point of $A$? – Brian M. Scott Nov 02 '13 at 06:43
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defnitely not.. – ILoveMath Nov 02 '13 at 06:46
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@DonAnselmo: So $x$ has an open nbhd disjoint from $A$. What must that nbhd be? – Brian M. Scott Nov 02 '13 at 06:46
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a singleton Brian – ILoveMath Nov 02 '13 at 06:47
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@DonAnselmo: Exactly. – Brian M. Scott Nov 02 '13 at 06:48