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Let $X $ be a compact non-Hausdorff topological space with the following property: for every infinite subset of closed points, say $\{x_i\}_{i \in I}$, there exists $j\in I$ such that $x_j$ is in the closure of $\{x_i\}_{i \not=j}$.

Is there any characterization of such spaces? Or is there any sufficient or necessary condition for this property? Thanks for any help.

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    $\begingroup$ You should edit your question to be clearer. $\endgroup$ – Chris Ramsey Dec 2 '15 at 17:26
  • $\begingroup$ I fixed up the grammar a little. To clarify, a "closed point" is a point $x$ such that $\{x\}$ is closed? $\endgroup$ – Nate Eldredge Dec 2 '15 at 17:32
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    $\begingroup$ Your edit reversed the changes I made; did you mean to do that? $\endgroup$ – Nate Eldredge Dec 2 '15 at 17:36
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    $\begingroup$ This looks rather like a homework exercise. What's the motivation for asking the question...? $\endgroup$ – Paul Levy Dec 2 '15 at 21:29

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