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 $U\subseteq X\times Y$ such that for every open set $W\subseteq Y$ 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.