What closure of a given subset really is? I don't understand what (intuitively) a closure of a closed set is. Can someone explain it to me?
There are many definitions around like the smallest closed set containing the set or the union of the set and it's limits point etc.
Do this all means that closure of a set= the union of the boundary points?
I would really apreciate a child's explanation using sequences.
Sorry for the format I'm new in ME.
Thanks in advance.
 A: Intuitively, and using sequences, the closure of a set is all the points you can reach by following a convergent sequence of points in the set. All the points in the set can be reached this way - just use constant sequences.
There may be points outside the set that you can reach with convergent sequences from inside: consider the open interval $(0,1)$ and the point $0$.
A set is closed just when all the points you can reach this way are already in the set. Any other definition of "closed" (using neighborhoods, or boundary points) can be proven equivalent to this one.
Your question asks about the "closure of a closed set". The closure of a closed set is the set itself.
A: I think the best way to think about closures (and a few other ideas in topology) is with the following proposition:

If $A$ is some subset of a topological space $X$, the interior, exterior, and boundary form a partition of $X$.

If you haven't seen these terms in a math context yet, that's fine. In order to build intuition here, you don't really need to know anything beyond their english meanings---the interior of $A$ is something vaguely "inside" $A$, the exterior is "outside" $A$, and the boundary is somewhere "in the middle". Our proposition says that every point in the topological space is in exactly one of the interior, exterior and boundary.
As an example, if $A = [0,1)$, the interior of $A$ is $(0,1)$, the exterior is $(-\infty, 0) \cup (1,\infty)$, and the boundary is $\{0,1\}$, which should agree with your intuition regarding real numbers, and what the words "inside," "outside," and "boundary" mean in english.

The closure of a set is simply the union of the interior and the boundary. This is a proposition, yes, but I like to think of it as an alternative definition. It gives you a nice intuitive---the closure of $A$ is just everything that isn't "outside" $A$, or equivalently everything that's "inside" and "on the boundary."
The fact that the closure of the set is related to sequences is a nice fact, but the intuition comes from the partition. 
