Limit of a sequence and a closed set 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?
 A: 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:


*

*When is an accumulation point not the limit of some sequence in a topological space?

*If $a\in \mathrm{clo}(S)$, does it follow that there exists a sequence of points in $S$ that converges to $a$?

*Limit points in topological space $X$

*Sequentially closed $\implies$ closed, but not Fréchet-Urysohn space

*Is a topology determined by its convergent sequences? at MathOverflow

*Example of different topologies with same convergent sequences

*is a net stronger than a transfinite sequence for characterizing topology?
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.
A: 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.  
