  This paper investigates the interrelationships between the observability
inequality, the H\"older-type interpolation inequality, and the spectral
inequality for the degenerate parabolic equation in $\mathbb{R}$. We elucidate
the distinctive properties of observable sets pertaining to the degenerate
parabolic equation. Specifically, we establish that a measurable set in
$\mathbb{R}$ fulfills the observability inequality when it exhibits
$\gamma$-thickness at a scale $L$, where $\gamma>0$ and $L>0$.
