Universal sets in metric spaces (I am cross-posting this from math.SE as it seems to be slightly over the top for that site.)
I saw in the class the theorem:
Suppose $X$ is a separable metric space, and $Y$ is a polish space (metric, separable and complete) then there exists a $G\subseteq X\times Y$ which is open and has the property:
For all $U\subseteq X$ open, there exists $y\in Y$ such that $U = \{x\mid\langle x,y\rangle\in G\}$.
$G$ with this property is called universal.
The proof is relatively simple, however the $y$ we have from it is far from unique, in fact it seems that it is almost immediate that there are countably many $y$'s with this property.
My question is whether or not this $G$ can be modified such that for every $U\subseteq X$ open there is a unique $y\in Y$ such that $U = \{x\mid\langle x,y\rangle\in G\}$?
Perhaps we need to require more, or possibly even less, from $X$ and $Y$?
Some thoughts:
Firstly $X$ cannot be finite, otherwise there are less than continuum many open subsets, and since $G$ is open we have that the projection on $Y$ is open, since $Y$ is Polish we have that this projection is of cardinality continuum, which in turn implies there are continuum many $y$'s with the same cut.
Secondly, as the usual proof goes through a Lusin scheme over $Y$, and using it to define $G$, I thought at first that using the axiom of choice we can select a set of points on which the mapping to open sets of $X$ is 1-1, and somehow remove some of the sets from the scheme. This proved to be a bad idea, as we remove sets that can be used for other open sets.
Thirdly, I thought about enumerating the open sets according to a rational enumeration so $A_i\subseteq A_j$ if and only if $q_i\le q_j$, and then instead of just placing the open sets of $X$ arbitrarily by the Lusin scheme, we use the rationals somehow.
 A: If $X =\omega$ with the discrete topology and $Y= \mathcal{P}(\omega)$ with the Cantor set topology 
let $G$ be the set of all $(A,n)$ such that $n\in A$.
A: While idly browsing around I stumbled over the follwing paper and remembered this question:
A.W. Miller, Uniquely Universal Sets, Topology and its Applications 159 (2012), pp. 3033–3041. It's available in various formats here.
Let me quote the abstract (to avoid confusion: Miller's terminology reverses the rôles of $X$ and $Y$ in your question):

We say that $X \times Y$ satisfies the Uniquely Universal property (UU)
  iff there exists an open set $U \subseteq X \times Y$ such that for
  every open set $W \subseteq Y$ there is a unique cross section of
  $U$ with $U_x=W$.  Michael Hrušák raised the question of when does
  $X \times Y$ satisfy UU and noted that if $Y$ is compact, then $X$
  must have an isolated point.  We consider the problem when
  the parameter space $X$ is either the Cantor space $2^\omega$ or the Baire
  space $\omega^\omega$.
  We prove the following:
  
  
*
  
*If $Y$ is a locally compact zero dimensional
  Polish space which is not compact, then $2^\omega\times Y$ has UU.
  
*If $Y$ is Polish, then
  $\omega^\omega \times Y$ has UU iff $Y$ is not compact.
  
*If $Y$ is a $\sigma$-compact subset of a Polish space which is
  not compact, then $\omega^\omega \times Y$ has UU.

His results are mostly positive: “a certain space or family of spaces has UU” and various permanence properties. One nice “negative” result:

Proposition 30: There exists a partition $X\cup Y=2^\omega$ into Bernstein sets
  $X$ and $Y$ such that for every Polish space $Z$ neither
  $Z\times X$ nor $Z\times Y$ has UU.

He also raises a few questions, e.g.:


*

*Question 4: Does $(2^\omega\oplus 1) \times [0,1]$ have UU?

*Question 6: Does either $\mathbb{R} \times \omega$
or $[0,1]\times \omega$ have UU?
Or more generally, is there any example of UU for a
connected parameter space?

*Question 11: Is the converse of Corollary 10 false?
That is: Does there exist $Y$ such that $\omega^\omega \times Y$ has UU but
$2^\omega\times Y$ does not have UU?

A: The possibly unsatisfying answer to your question is "sometimes."  I will instead discuss the obviously equivalent question about universal closed subsets (it will let me use more standard notation later).  Moreover, I will focus on the special case that $X$ and $Y$ are both Polish, since that has been examined more in the literature.
First, let me point out an oversight in your analysis of the case that $X$ is finite.  Certainly $X$ must have the discrete topology, so every subset of $X$ is closed.  However, $Y = \mathcal{P}(X)$ is a perfectly fine Polish space when endowed with its own discrete topology.  Then the set $\{(x,A) \in X \times \mathcal{P}(X) : x \in A\}$ is "uniquely" universal closed.
This may seem pedantic, but it actually generalizes to large $X$.  Suppose now that $X$ is a compact Polish space, and endow its space of compact (equiv., closed) subsets $\mathcal{K}(X)$ with the Vietoris topology, generated by sets of the form
$\{K : K \subseteq U\}$ and $\{K : K \cap U = \emptyset\}$,
where $U\subseteq X$ is open.  For Polish $X$, this is a Polish topology on $\mathcal{K}(X)$.  Note that in the special case where $X$ is finite (thus compact), this coincides with the discrete topology on $\mathcal{P}(X)$.  Motivated by this analogy, we proceed as before and choose our uniquely universal closed set to equal $G = \{(x,K) \in X \times \mathcal{K}(X) : x \in K\}$.  The only thing left to check is that this set is indeed closed.  You can see this directly by assuming $(x_0,K_0) \notin G$, fixing a little open neighborhood $U$ around $x_0$ disjoint from $K_0$, and then checking that $U \times \{K : K \cap U = \emptyset\}$ is an open neighborhood of $(x_0, K_0)$ disjoint from $G$.
The obvious place to look for more information about this is Kechris' descriptive set theory text.  Unfortunately I don't have a copy on hand at the moment (which makes me feel like a child without a security blanket), so I can't give more specific references.
Moving on.  For noncompact Polish spaces $X$ you can endow the space $\mathrm{CL}(X)$ of closed subsets of $X$ with a topology called the Wijsman topology.  Well, really there are several such topologies, since the definition relies on a choice of compatible complete metric $d$ on $X$.  This topology is the weakest topology making the functions $f_x : A \mapsto d(x, A)$ continuous for each $x \in X$.  It is a result of Gerald Beer's that this topology is Polish for $(X,d)$ as above.  (This might well be in Kechris' book, but as I mentioned I don't have it on hand so I'll regurgitate the reference that google gave me.)
Beer, Gerald. A Polish topology for the closed subsets of a Polish space. Proc. Amer. Math. Soc. 113 (1991), no. 4, 1123–1133. 
Edit: but Theo Buehler has given a relevant reference to Kechris.  See his comment.
A variation of the earlier argument in the compact case should work in this context.
Edit again: I just noticed that the definition of the topology I gave makes sense for nonempty closed subsets of $X$.  This is not a serious problem and is in fact addressed in Beer's paper.
Finally, it is hopeless to expect this to work for arbitrary Polish spaces $X$ and $Y$.  As you noticed, for small spaces there are cardinality issues.  When the spaces are large, you can also fiddle around with compactness/noncompactness, and other topological notions.  There are just too many wild Polish spaces.
