nLab
diagonal function

Given a set $X$, its diagonal function is a function from $X$ to its cartesian square $X^2$, often denoted ${\Delta }_X$, $\check{X}$, or an obvious variation.

Specifically, the diagonal function of $X$ maps an element $a$ of $X$ to the pair $(a,a)$:

$${\Delta }_X=\{a\mapsto (a,a)\}.$$\Delta_X = \{ a \mapsto (a,a) \} .

Note that this map is an injection, so it defines a subset of $X^2$, also called the diagonal of $X$; this is the origin of the term.

The concept can be generalised to any category in which the product $X^2$ exists; see diagonal morphism.

A topological space $X$ is Hausdorff if and only if its diagonal function is a closed map; this fact can be generalised to other notions of space.