@MISC{WEIERSTR.MIZ,
  AUTHOR = {Bia{\l}as, J\'ozef and Nakamura, Yatsuka},
  TITLE = {The Theorem of {W}eierstrass},
  SECTION1 = {Preliminaries},
  SECTION2 = {The Weierstrass' Theorem},
  SECTION3 = {The Measure of the Distance Between Compact Sets},
  DAY = {10},
  MONTH = {July},
  YEAR = {1995},
  ADDRESS1 = {{\L}\'od\'z \ University\\{\L}\'od\'z},
  ADDRESS2 = {Shinshu University\\Nagano},
  SUMMARY = {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 $x_1$, $x_2$ of $T$ being elements of $S$, 
    such
    that $f(x_{1})$ and $f(x_{2})$ 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.}}

