Introduction to the Mathematical and Statistical Foundations of Econometrics

Sets in Euclidean Spaces

An open e-neighborhood of a point x in a Euclidean space Kk is a set of the form

Ne(x) = {y e Kk : Уy — x У < e}, e > 0, and a closed e-neighborhood is a set of the form

Ne(x) = {y e Kk : Уy — x У < e}, e> 0.

A set A is considered open if for every x e A there exists a small open e-neighborhood Ne(x) contained in A. In shorthand notation: Vx e A 3e > 0: Ne(x) c A, where V stands for “for all” and 3 stands for “there exists.” Note that the e’s may be different for different x.

A point x is called a point of closure of a subset A of Kk if every open e - neighborhood Ne (x) contains a point in A as well as a point in the complement A of A. Note that points of closure may not exist, and if one exists it may not be contained in A. For example, the Euclidean space Kk itself has no points of closure because its complement is empty. Moreover, the interval (0,1) has two points of closure, 0 and 1, both not included in (0,1). The boundary of a set A, denoted by d A, is the set of points of closure of A. Again, d A may be empty. A set A is closed if it contains all its points of closure provided they exist. In other words, A is closed if and only if d A =0 and d A c A. Similarly, a set A is open if either d A = 0 or d A c A. The closure of a set A, denoted by A, is the union of A and its boundary d A: A = A U d A. The set Ad A is the interior of A.

Finally, if for each pair x, y of points in a set A and an arbitrary X e [0, 1] the convex combination z = Xx + (1 — X)y is also a point in A, then the set A is called convex.