Binary Space (B,d) is a metric space as follows. Let BB be the set of all infinite sequences of 0's and 1's. That is, points of B are of the form x=(x1,x2,x3,…), where xi=0 or 1 for every i=1,2,3,…. If x=(x1,x2,x3,…),y=(y1,y2,y3,…)belong to B, define d(x,y)=0 if x=y; otherwise 1/n if x doesn't equal to y and n=min{i∣xi≠yi}.
How can I describe the set of all limit points of B?
I'm going to assume you proved that $d$ is a metric in $B$, if not, the question doesn't makes sense. The set of all limit points of $B$ is $B$ itself. To see this, take any point $x$ in $B$, $x=(x_1,x_2,\ldots)$, you can construct a sequence ${x^k}$ of points in $B$ whose limit is $x$. Just define the point $x^k$ to be equal to $x$ at least up to the $k$-th term, i.e $x^k_i=x_i$ $\forall$ $i\leq k$. Then $d(x^k,x) \leq 1/k$. Hence any point is a limit point in $B$.