Let [itex]A\subset X[/itex] be a subset of some topological space. If [itex]x\in\overline{A}\backslash A[/itex], does there exist a sequence [itex]x_n\in A[/itex] so that [itex]x_n\to x[/itex]?(adsbygoogle = window.adsbygoogle || []).push({});

In fact I already believe, that such sequence does not exist in general, but I'm just making sure. Is there any standard counter examples? I haven't seen any.

If the point x has a countable local basis, that means a set [itex]\{U_n\in\mathcal{T}\;|\;n\in\mathbb{N}\}[/itex] so that [itex]x\in U_n[/itex] for all n, that for all environments U of x, there exists some [itex]U_n\subset U[/itex], then the sequence [itex]x_n[/itex] is simple to construct.

I guess that the existence of the countable local basis is a nontrivial matter, and since the sequence comes so naturally with it, it seems natural to assume that the sequence would not exist without the countable local basis. But what kind of topological spaces don't have countable local basis for each point? And what would be an example of a point in closure, that cannot be approached by some sequence?

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# Limit in closure, topology stuff

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