Parabola obstruction to countable line coverings
= Parabola obstruction to countable line coverings
A <parabola> parametrized by $(t,t^2)$ is in <bijection> with $\mathbb R$ and is an <uncountable set>. Every affine line meets it in at most two points. A countable family of affine lines could therefore cover only countably many points of this <parabola>, and cannot cover the plane. More generally, an uncountable curve with finite intersection with every permitted member obstructs a countable covering by those members.