Let $X$ be a reduced scheme. If there are arbitrarily chosen two points $p_1, p_2 \in X$, does the following always hold?
Q. There exists some affine open neighbourhood $U$ such that $U \ni p_1, p_2$.
I know this holds when ${\mathrm{dim}}. X = 1$, and when $p_i$ are generic points of divisors, or irreducible components, still holds.
However, I have difficulty to prove this when $p_i$ are maximal ideals, i.e., points.