The Theorem of Weierstrass

The basic purpose of this article is to prove the important Weierstrass’ 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 infimum, respectively, of f(S), which is the image of S under the function f . The paper is divided into three parts. In the first part, we prove some auxiliary theorems concerning properties of balls in metric spaces and define 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 final part of this work is developed for the purpose of defining some measures of the distance between compact subsets of topological metric spaces. Some simple theorems about these measures are also proved.