Correspondences

We will call a point-to-set mapping a correspondence. A function maps points into points. A correspondence (or point-to-set mapping) maps points into sets of points. Let A and B be sets. We would like to describe a correspondence from A to B. For each x ∈ A we associate a nonempty set β ⊂ B by a rule ϕ. Then we say β = ϕ(x), and ϕ is a correspondence. The notation to designate this mapping is ϕ : A → B. For example, suppose A and B are both the set of human population. Then we could let ϕ be the cousin correspondence ϕ(x) = {y | y is x's cousin}. Note that if x ∈ A and y ∈ B, it is meaningless or false to say y = ϕ(x), rather we say y ∈ ϕ(x). The graph of the correspondence is a subset of A × B : {(x, y) | x ∈ A, y ∈ B and y ∈ ϕ(x)}.

For example, let A = B = R. We might consider ϕ(x) = {y | x - 1 ≤ y ≤ x + 1}. The graph of ϕ(·) appears in Figure 23.1.

Upper hemicontinuity (also known as upper semicontinuity)

In the balance of this chapter and the next, we concentrate on mappings from one real Euclidean space into another, from RN into RK, for N ≥ 1 and K ≥ 1.