If $S \subset X$ does a subsequence of $S$ converge in $S$ or in $X$?

Asked May 27 '16 at 16:39

Active May 27 '16 at 16:39

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Let $X$ be a metric space and let $S\subset X$ be a compact space.

By definition, $S$ is compact implies that for all sequences $(x_n)$ of $S$, there exists a subsequence $(x_{n_{\alpha}})$ that converges.

My question is Does $(x_{n_{\alpha}})$ only converge in S or can it converge in X?

asked May 27 '16 at 16:39

aribaldi

2

By definition, $S$ is compact implies that for all sequences $(x_n)$ of $S$, there exists a subsequence $(x_{n_{\alpha}})$ that converges and whose limit lies in S. – Sri-Amirthan Theivendran May 27 '16 at 16:40
what if S is open in X? – aribaldi May 27 '16 at 16:42
@aribaldi: That makes no difference at all. – Brian M. Scott May 27 '16 at 16:44

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