# 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][1]:

• [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]

In topology: