  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.
