Examples I could think of are all sequences with their limit. But is every countably infinite compact space admit atleast one isolated point?
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What about $\Bbb N$ with the cofinite topology (open sets are those with finite complement)? – David Mitra Jun 11 '15 at 14:35
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...and the empty set, of course. – David Mitra Jun 11 '15 at 14:44
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Instead of topological space, what if we have a metric space? – Mambo Jun 11 '15 at 14:59
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Hausdorff is enough. See this. – David Mitra Jun 11 '15 at 15:09
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@DavidMitra Exactly. – Mambo Jun 12 '15 at 08:23
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Let $(X, \mathcal T)$ be a compact topological space (countable or not, it does not matter). For $x \notin X$ let $Y = X \cup \{x\}$ be a new topological space with the topology generated by $\mathcal T \cup \{ \{x\} \}$.
Then $Y$ is compact and $x$ is an isolated point in $Y$.
Alex M.
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