@MISC{TSP_2.MIZ,
  AUTHOR = {Karno, Zbigniew},
  TITLE = {Maximal {K}olmogorov Subspaces of a Topological Space as {S}tone Retracts of the Ambient Space},
  NOTE = {Presented at Mizar Conference: Mathematics in Mizar (Bia{\l}ystok, September 
 12--14, 1994).},
  SECTION1 = {Maximal $T_{0}$-Subsets},
  SECTION2 = {Maximal Kolmogorov Subspaces},
  SECTION3 = {Stone Retraction Mapping Theorems},
  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
(see \cite{ALEX-HOPF}, \cite{KURAT:2}, \cite{ENGEL:1}). Such spaces
and subspaces were investigated in Mizar formalism in \cite{TSP_1.ABS}.
A Kolmogorov subspace $X_{0}$ of a topological space $X$ is said to be
{\em maximal}\/ provided for every Kolmogorov subspace $Y$ of $X$
if $X_{0}$ is subspace of $Y$ then the topological structures of
 $Y$ and $X_{0}$ are the same.\par
 M.H.~Stone proved in \cite{STONE:3} 
that every topological space
can be made into a Kolmogorov space by identifying points with the same
closure (see also \cite{THRON:1}). The purpose is to generalize the
Stone result, using Mizar System. It is shown here that: (1) {\em in every
topological space $X$ there exists a maximal Kolmogorov subspace $X_{0}$
 of $X$}, and (2) {\em every maximal Kolmogorov subspace $X_{0}$ of
 $X$ is a continuous retract of $X$}.
Moreover, {\em if $r : X \rightarrow X_{0}$ is a continuous
retraction of $X$ onto a maximal Kolmogorov subspace $X_{0}$
of $X$, then $r^{-1}(x) = {\rm MaxADSet}(x)$ for any point $x$ of $X$
 belonging to $X_{0}$, where ${\rm MaxADSet}(x)$ is a unique maximal
anti-discrete subset of $X$ containing $x$} (see \cite{TEX_4.ABS}
for the precise definition of the set ${\rm MaxADSet}(x)$).
The retraction $r$ from the last theorem is defined uniquely, and it is denoted
in the text by ``Stone-retraction". It has the following two remarkable
properties: $r$ is open, i.e., sends open sets in $X$ to open sets in
 $X_{0}$, and $r$ is closed, i.e., sends closed sets in $X$ to closed sets
in $X_{0}$.\par
These results may be obtained by the methods described by R.H. Warren
in \cite{WARREN:1}.}}


@BOOK{THRON:1,
       AUTHOR = {W.J. Thron},
        TITLE = {Topological Structures},
    PUBLISHER = {Holt, Rinehart and Winston},
         YEAR = {1966},
      ADDRESS = {New York}}

@ARTICLE{WARREN:1,
       AUTHOR = {R.H. Warren},
        TITLE = {Identification spaces and unique uniformity},
      JOURNAL = {Pacific Journal of Mathematics},
         YEAR = {1981},
       VOLUME = {95},
        PAGES = {483--492}}

