Let $X$ be metrizable and let $\langle x_{n}\rangle$ be sequence in $X$. Can we say the set of sequence closed discrete? How can I define a closed discrete subset by using sequence? Could you give me any idea?
construction closed discrete set by using sequence
2 Answers
No, for instance if $X=\mathbb R$ and $\{x_n\}$ is an enumeration of the rationals. Then the set of the sequence is neither closed nor discrete. If you want to define a closed discrete subset by using a sequence you need to pick a sequence whose only convergent subsequences are eventually constant. In a non compact metric space $X$ we can find a sequence without a convergent subsequence, by the failure of sequential compactness. This would give rise to a closed discrete subspace, because it has no limit points.
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0oh sorry, it is clear. thanks so much. – 2012-11-28
A subset $A$ of $X$ is closed and discrete if and only if it hasn't an accumulation.
Proof: Let $A$ be a closed discret subset of $X$. Obviously, the points of $A$ are all discrete, which cannot be the accumulation of $A$.
Let $A$ hasn't an accumulation, then $A$ is closed. Moreover, every point of $A$ is not the accumulation point of $A$, therefore, it must be discrete.