The diagonal (set)

Definition

Let $X$ be a set and let $X\times X$ denote the 2-fold Cartesian product of $X$ with itself, we call the following set the diagonal of $X$ in $X\times X$:

• $D:\eq\{(x,x)\in X\times X\ \vert\ x\in X\}$
  • Here $(x,x)$ denotes the ordered pair whose first element is $x$ and whose second element is also $x$

Uses

In topology:

• A topological space $(X,\mathcal{J})$ is Hausdorff $\iff$ the diagonal $D\subseteq X\times X$ is closed in $X\times X$ (considered with its product topology)