Research proof index

At every point $p\in\mathbb C$, the axis-parallel closed squares centered at $p$ with positive half-width form a neighborhood basis. Equivalently, a set is a neighborhood of $p$ exactly when it contains one of these sufficiently small squares.

Let $R$ be the axis-parallel rectangle with opposite corners $z,w\in\mathbb C$. If a point $p$ lies strictly to the left, right, below, or above both corners, then $p\notin R$; equivalently, the rectangle and the singleton $\{p\}$ are disjoint.