# nLab diagonal subset

Given a set $X$, its diagonal is a subset of its cartesian square ${X}^{2}$, often denoted ${\Delta }_{X}$, ${I}_{X}$, or an obvious variation.

Specifically, the diagonal of $X$ consists of those pairs of the form $\left(a,a\right)$ for $a$ an element of $X$:

${\Delta }_{X}=\left\{\left(a,a\right)\mid a\in X\right\}.$\Delta_X = \{ (a,a) | a \in X \} .

The term “diagonal” arises because if we arrange the elements of ${X}^{2}$ in a matrix with the rows and columns labeled by elements of $X$ in the same order, then ${\Delta }_{X}$ consists precisely of entries along the diagonal of the matrix.

When intepreted as a binary relation, ${\Delta }_{X}$ is the equality relation on $X$. This relation is both functional and entire; when interpreted as a function, it is the identity function on $X$. Note that there is an obvious bijection $a↦\left(a,a\right)$ from $X$ to ${\Delta }_{X}$; thus, we can also interpret the diagonal as a function from $X$ to ${X}^{2}$, called the diagonal function.

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

A topological space $X$ is Hausdorff if and only if its diagonal is a closed subspace of ${X}^{2}$; this fact can be generalised to other notions of space.

Revised on August 26, 2012 10:22:27 by Toby Bartels (98.19.40.130)