Uniquely universal sets in $\mathbb{R}\times \omega$ and $[0,1]\times \omega$

Let $X$ and $Y$ be topological spaces. We say that $X\times Y$ satisfies the Uniquely Universal property (UU) iff there exists an open set $U\subseteq X\times Y$suchthatforeveryopenset$W\subseteq Y$ there is a unique cross section of $U$ with $U\left( x\right) =\left{ y\in Y:\left( x,y\right) \in U\right} =W$. Arnold W. Miller in his paper \cite{1} posed the following two questions:

1. Does $[0,1]\times \omega$ have UU?
2. Does $\mathbb{R}\times \omega$ have UU? In this paper we present two constructions which give positive answers to both problems.