The diagonal (set)

Definition

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

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

Uses

In topology:

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