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It's a dumb question, but I need to assure myself:

If $V$ is a subset of a metric space $W$, then if we take a sequence in $V$ and it has a limit in $W\setminus V$, does it mean that $V$ is not closed? What if $W$ is not a metric space?

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up vote 3 down vote accepted

In a metric space a set is closed iff it contains all it's sequential limits. In a topological space every closed set contains it's sequential limits but there are sets containing all their sequential limits which are not closed.

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Thank you. Can you think of any example of those sets? – luka5z Apr 19 '14 at 15:56
They are pathological because they can't be first countable. One example in Munkres is the following: $A$ is a well ordered set with largest element $\Omega$ where the section $S_\Omega$ is uncountable but every other section is countable. Then $S_\Omega$ is not closed but it contains all it's sequential limits. See Munkes for details. – Seth Apr 19 '14 at 16:04

This is more like a longer comment than an answer, but it is probably more readable than a series of shorter comments. (And also easier to correct, if I mix up something.)

Topological spaces, in which closed sets are precisely the sets closed under taking limits of sequences, are called sequential spaces. (I.e. a subset is closed if and only if it is sequentially closed.)

Spaces, in which closure of any given subset can be obtained by taking limits of all convergent sequences of elements of this set, are called Frechet-Urysohn spaces.

I will mention at least the basic relation between these spaces:
metric space $\Rightarrow$ first-countable space $\Rightarrow$ Frechet-Urysohn space $\Rightarrow$ sequential space
(Neither of these implications can be reversed.)

So you are basically asking for an examples of spaces, which are not sequential and examples of spaces which are not Frechet-Urysohn.

There are already several post, which give such examples:

In connection with sequential spaces, a series of articles on Dan Ma's topology blog is worth mentioning.

Probably any decent text which introduces nets in topological spaces would also have an example showing that sequences are not sufficient to describe topology completely.

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Beautiful. Thanks a lot! – luka5z Apr 19 '14 at 17:58
I would change your definition of Frechet space to be "any given subset" to be more clear. The important thing to note here is that sequential closure is not idempotent in a general topological space so a sequential space need not be Frechet. This was a point of confusion for me. – Seth Apr 19 '14 at 22:13
@Seth I've changed the wording. – Martin Sleziak Apr 20 '14 at 6:45

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