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My question is: what are the minimal conditions on a topological space for it have the following property?

$$x\in \bar{A}\iff \exists (x_n)\subset A | x_n \to x$$

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    I assume you require $x\notin \{x_n\}$?2012-03-28
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    Not necessarily! I am not considering only accumulation points!2012-03-28
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    http://en.wikipedia.org/wiki/Sequential_space2012-03-28
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    @AndréLima A space with that property is called a Frechet space. (@t.b.) there are sequential spaces that are not Frechet a classical example is the sequential fan or Aren's space.2012-03-28
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    @azarel: thanks for the clarification, I thought the page I linked to was enough (there's a section on Fréchet-Urysohn spaces there).2012-03-28
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    @t.b. My bad, I just read the first paragraph on the link.2012-03-28

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