Pointwise - Pointwise Relations

Pointwise Relations

In order theory it is common to define a pointwise partial order on functions. With A, B posets, the set of functions A → B can be ordered by f ≤ g if and only if (∀x ∈ A) f(x) ≤ g(x). Pointwise orders also inherit some properties of the underlying posets. For instance if A and B are continuous lattices, then so is the set of functions A → B with pointwise order. Using the pointwise order on functions one can concisely define other important notions, for instance:

• A closure operator c on a poset P is a monotone and idempotent self-map on P (i.e. a projection operator) with the additional property that id_A ≤ c, where id is the identity function.

• Similarly, a projection operator k is called a kernel operator if and only if k ≤ id_A.

An example of infinitary pointwise relation is pointwise convergence of functions — a sequence of functions

with

converges pointwise to a function if for each in