diagonal subset

Given a set XX, its diagonal is a subset of its cartesian square X 2X^2, often denoted Δ X\Delta_X, I XI_X, or an obvious variation.

Specifically, the diagonal of XX consists of those pairs of the form (a,a)(a,a) for aa an element of XX:

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

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

When intepreted as a binary relation, Δ X\Delta_X is the equality relation on XX. This relation is both functional and entire; when interpreted as a function, it is the identity function on XX. Note that there is an obvious bijection a(a,a)a \mapsto (a,a) from XX to Δ X\Delta_X; thus, we can also interpret the diagonal as a function from XX to X 2X^2, called the diagonal function.

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

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

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

Last revised on August 15, 2014 at 09:15:59. See the history of this page for a list of all contributions to it.