1. The problem is to understand the Cartesian product of two intervals $[a,b]$ and $[0,1]$. 2. The Cartesian product $[a,b] \times [0,1]$ represents all ordered pairs $(x,y)$ where $x$ is in $[a,b]$ and $y$ is in $[0,1]$. 3. This means $x$ satisfies $a \leq x \leq b$ and $y$ satisfies $0 \leq y \leq 1$. 4. Geometrically, this forms a rectangle in the coordinate plane with horizontal side from $a$ to $b$ and vertical side from $0$ to $1$. 5. There is no further simplification; the set is exactly all points $(x,y)$ with $x \in [a,b]$ and $y \in [0,1]$. 6. This concept is fundamental in analysis and topology when dealing with product spaces and intervals.