Convergence in distribution-topological interpretation There are various definitions of convergence of measurable functions: some of them have clear topological interpretation (as coming from suitable norm or metric-for example convergence in $L^p$ or convergence in probability) while some of them not (convergence almost surely does not come from any topology). I would like to understand the notion of convergence in distribution from the point of view of topology. Some properties which shows that one has to be careful are as follows:
-one can have two different random variables $X,Y$ which have the same distributions: thus the sequence $X,Y,X,Y,X,Y,...$ would be convergent and have two limits $X,Y$-this contradicts Hausdorff property (or even being $T_1$)
-one can have even two different measure spaces $(\Omega_1,\mathcal{F}_1,P_1),(\Omega_2,\mathcal{F}_2,P_2)$ on which $X$ and $Y$ are defined and still they can have the same distribution. Therefore it is not clear on which space should those random variables act (should it be fixed or not?).
So to summarize:  

Does convergence in distribution comes from some topology (Hausdorff topology?) and if the answer is ,,yes'' what is the underlying set on which this topology is defined?

 A: The answer is "YES"! 
But let us first make it clear - just as Giuseppe Negro pointed out -that convergence in distribution is more about measures then random variables. Even if
$$X_{n}\ \xrightarrow{d} X$$
and all $X_{n} \ $, and $X$ are defined on common space $(\Omega, \mathcal{F}, \mathbb{P}), \ $ we still can't tell anything about a sequence
$$X_{1}(\omega), X_{2}(\omega),\cdots, X_{n}(\omega),\cdots$$ 
for a fixed $\ \omega. $ Convergence in distribution does not say anything about concrete realizations of $X_{n}$, $X$.  Hence

one can have two different random variables $X,Y$ which have the same distributions: thus the sequence $X,Y,X,Y,X,Y,\cdots$ would be convergent and have two limits $X,Y$. 

is not really the case. 

Does convergence in distribution comes from some topology (Hausdorff topology?) and if the answer is ,,yes'' what is the underlying set on which this topology is defined?

In case of random variables - measure (distribution) is uniquely defined by their CDF. For any two CDF functions $\ F,G \ $ corresponding to some probability measures on $ \ (\mathbb{R},\mathcal{B}(\mathbb{R})) \ $, we define 
$$d(F,G)=\inf \{\epsilon\ge 0: \forall_{x\in \mathbb{R}} \ G(x-\epsilon)-\epsilon \le F(x)\le G(x+\epsilon) +\epsilon\}.$$
This is called Levy metric. It can be in fact shown that:
a) it is a true metric on set of all probability  measures on $ \ (\mathbb{R},\mathcal{B}(\mathbb{R})) \ $.
b) $\ F_{n}\xrightarrow{d} F \ \iff \ d(F_{n},F)\rightarrow 0.$
