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

TL;DR

This paper proves that the spaces [0,1]×ω and ℝ×ω possess the Uniquely Universal property by providing explicit constructions, answering longstanding open questions in topological product spaces.

Contribution

The authors construct explicit examples demonstrating that both [0,1]×ω and ℝ×ω satisfy the Uniquely Universal property, resolving two open problems posed by Arnold W. Miller.

Findings

Both spaces have the UU property.

Explicit constructions are provided for each space.

Answers two open questions in topology.

Abstract

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.