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 $(a,a)$ for $a$ an element of $X$:

$$\Delta_X = \{ (a,a) | a \in X \} .$$

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 $a \mapsto (a,a)$ 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.

The characteristic function of the diagonal subset is the Kronecker delta.