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It is known that if a space $X $ is metricable then for any subset $A $ of $X $ and a point $x \in \overline A $ there is a sequence of points in $A $ converging to $x $.

I wonder if there is an example of a topological space $X $ (not metricable then) where this doesn't always hold?

Thanks in advance!

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  • $\begingroup$ Are you requiring $x\in \overline{A}\setminus A$ (not stated as such)? Are you allowing the sequence to consist of $x_n=\{x,x,x,\dots\}$? $\endgroup$ – JohnD Nov 25 '14 at 17:12
  • $\begingroup$ @JohnD: wouldn't you allow these? In a discrete space, it is necessary to have those in order to have equivalence between $\overline A$ and limits of sequences. It would be silly to discard this type of sequence, and withdraw the generality of the statement "In any metrizable space X, ...". $\endgroup$ – zarathustra Nov 25 '14 at 17:18
  • $\begingroup$ Yes, I had your example in mind below ;-). I was just trying to reconcile the statement in the question title with the statement in the question body. $\endgroup$ – JohnD Nov 25 '14 at 17:23
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There are many such spaces. One simple Hausdorff example is the Arens-Fort space, which is fairly clearly described in the linked article. In that space let $A$ be everything except the point $\langle 0,0\rangle$; then $\langle 0,0\rangle\in\operatorname{cl}A$, but no sequence in $A$ converges to $X$.

An even simpler example, but one that isn’t Hausdorff, is an uncountable set $X$ with the co-countable topology: the open sets are $\varnothing$ and the complements of countable sets. Let $p\in X$ be arbitrary, and let $A=X\setminus\{p\}$; then $p\in\operatorname{cl}A$, but no sequence in $A$ converges to $p$. (In fact, the only convergent sequences in $X$ are the trivial ones, i.e., the ones that are constant from some term on.)

There are also nice spaces with this property, e.g., compact Hausdorff spaces. If we let $X$ be the set of all ordinal numbers less than or equal to $\omega_1$, the first uncountable ordinal, and give $X$ the order topology, then $X$ is a compact Hausdorff space, $\omega_1\in\operatorname{cl}[0,\omega_1)$, and no sequence in $[0,\omega_1)$ converges to $\omega_1$.

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