@MISC{URYSOHN1.MIZ,
  AUTHOR = {Bia{\l}as, J\'ozef and Nakamura, Yatsuka},
  TITLE = {Dyadic Numbers and {T}${}_4$ Topological Spaces},
  SECTION1 = {Dyadic Numbers},
  SECTION2 = {Normal Spaces},
  SECTION3 = {Some Increasing Family of Sets in Normal Space},
  DAY = {29},
  MONTH = {July},
  YEAR = {1995},
  ADDRESS1 = {{\L}\'od\'z \ University\\{\L}\'od\'z},
  ADDRESS2 = {Shinshu University\\Nagano},
  SUMMARY = {This article is the first part of a paper proving
    the fundamental Urysohn's Theorem concerning the existence of
    a real valued continuous function on a normal topological space.
         The paper is divided into four parts. In the first part,
    we prove some auxiliary theorems concerning properties of
    natural numbers and prove two useful schemes about recurrently
    defined functions; in the second part, we define a special
    set of rational numbers, which we call dyadic, and prove some
    of its properties. The next part of the paper contains the
    definitions of T${}_1$ space and normal space, and we prove
    related theorems used in later parts of the paper. The final
    part of this work is developed for proving the theorem about
    the existence of some special family of subsets of a
    topological space. This theorem is essential in proving
    Urysohn's Lemma.}}

