The Theorem of Weierstrass

The basic purpose of this article is to prove the important Weier-strass' theorem which states that a real valued continuous function f on a topological space T assumes a maximum and a minimum value on the compact subset S of T , i.e., there exist points x1, x2 of T being elements of S, such that f(x1) and f(x2) are the supremum and the innmum, respectively, of f(S), which is the image of S under the function f. The paper is divided into three parts. In the rst part, we prove some auxiliary theorems concerning properties of balls in metric spaces and deene special families of subsets of topological spaces. These concepts are used in the next part of the paper which contains the essential part of the article, namely the formalization of the proof of Weierstrass' theorem. Here, we also prove a theorem concerning the compactness of images of compact sets of T under a continuous function. The nal part of this work is developed for the purpose of deening some measures of the distance between compact subsets of topological metric spaces. Some simple theorems about these measures are also proved. The following three propositions are true: (1) Let M be a metric space, x 1 , x 2 be points of M, and r 1 , r 2 be real numbers. Then there exists a point x of M and there exists a real number r such that Ball(x 1 ; r 1) Ball(x 2 ; r 2) Ball(x; r): (2) Let M be a metric space, n be a natural number, F be a family of subsets of M, and p be a nite sequence. Suppose F is nite and a family of balls and rng p = F and dom p = Seg(n + 1): Then there exists a family G of subsets of M such that (i) G is nite and a family of balls, and (ii) there exists a nite sequence q such that rng q = G and dom q = Seg n and there exists a point x of M and there exists a real number r such that S F S GBall(x; r): (3) Let M be a metric space and F be a family of subsets of M. Suppose F is nite and a family of balls. Then there exists a point x of M and there exists a real …