nLab
diagonal function

Given a set XX, its diagonal function is a function from XX to its cartesian square X 2X^2, often denoted Δ X\Delta_X, Xˇ\check{X}, or an obvious variation.

Specifically, the diagonal function of XX maps an element aa of XX to the pair (a,a)(a,a):

Δ X={a(a,a)}. \Delta_X = \{ a \mapsto (a,a) \} .

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

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

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

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

Revised on August 15, 2014 09:21:42 by Colin Zwanziger (50.198.99.123)