@MISC{TSP_1.MIZ,
  AUTHOR = {Karno, Zbigniew},
  TITLE = {On {K}olmogorov Topological Spaces},
  NOTE = {Presented at Mizar Conference: Mathematics in Mizar (Bia{\l}ystok, September
 12--14, 1994).},
  SECTION1 = {Subspaces},
  SECTION2 = {Kolmogorov Spaces},
  SECTION3 = { $T_{0}$-Subsets},
  SECTION4 = {Kolmogorov Subspaces},
  DAY = {26},
  MONTH = {July},
  YEAR = {1994},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  SUMMARY = {Let $X$ be a topological space.
 $X$ is said to be {\em $T_{0}$-space}\/ (or {\em Kolmogorov space})
provided for every pair of distinct points $x,\,y \in X$ there exists
an open subset of $X$ containing exactly one of these points;
equivalently, for every pair of distinct points $x,\,y \in X$ there exists
a closed subset of $X$ containing exactly one of these points
(see \cite{ALEX-HOPF}, \cite{KURAT:2}, \cite{ENGEL:1}).\par
 The purpose is to list some of the standard facts on Kolmogorov spaces,
 using Mizar formalism.
 As a sample we formulate the following characteristics of such spaces:
 {\em $X$ is a Kolmogorov space iff for every pair of distinct
 points $x,\,y \in X$ the closures $\overline{\{x\}}$ and
 $\overline{\{y\}}$ are distinct}.\par
  There is also reviewed analogous facts on Kolmogorov subspaces of topological
 spaces. In the presented approach $T_{0}$-subsets 
are introduced and some of their properties developed.}}


@BOOK{ALEX-HOPF,
       AUTHOR = {P. Alexandroff and H. H. Hopf},
        TITLE = {Topologie I},
    PUBLISHER = {Springer-Verlag},
         YEAR = {1935},
      ADDRESS = {Berlin}}