@MISC{TEX_2.MIZ,
  AUTHOR = {Karno, Zbigniew},
  TITLE = {Maximal Discrete Subspaces of Almost Discrete Topological Spaces},
  ACKNOWLEDGEMENT = {The author wishes to thank Professor
  A. Trybulec for many helpful conversations during the preparation of
  this paper. The author is also grateful to G.~Bancerek
  for the definition of the clustered attribute {\em proper}.},
  SECTION1 = {Proper Subsets of 1-sorted Structures},
  SECTION2 = {Proper Subspaces of Topological Spaces},
  SECTION3 = {Maximal Discrete Subsets and Subspaces},
  SECTION4 = {Maximal Discrete Subspaces of Almost Discrete Spaces},
  SECTION5 = {Continuous Mappings and Almost Discrete Spaces},
  DAY = {05},
  MONTH = {November},
  YEAR = {1993},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  SUMMARY = {Let $X$ be a topological space and let $D$ be a subset of $X$.
 $D$ is said to be {\em discrete}\/ provided for every subset
 $A$ of $X$ such that $A \subseteq D$ there is an open
subset $G$ of $X$ such that $A = D \cap G$\/
(comp. e.g., \cite{KURAT:2}).  A discrete subset $M$ of $X$ is
said to be {\em maximal discrete}\/ provided for every discrete
subset $D$ of $X$ if $M \subseteq D$ then $M = D$.
A subspace of $X$ is {\em discrete}\/ ({\em maximal discrete})
iff its carrier is discrete (maximal discrete) in $X$.\par
 Our purpose is to list a number of properties of
discrete and maximal discrete sets
in Mizar formalism. In particular, we show here that {\em if $D$
is dense and discrete then $D$ is maximal discrete}; moreover,
 {\em if $D$ is open and maximal discrete then $D$ is dense}.
We discuss also the problem of the existence of maximal discrete
subsets in a topological space.\par
 To present the main results we first recall a definition of a class
of topological spaces considered herein. A topological space $X$ is
called {\em almost discrete}\/ if every open subset of $X$ is closed;
equivalently, if every closed subset of $X$ is open.
Such spaces were investigated in Mizar formalism in
 \cite{TDLAT_3.ABS} and \cite{TEX_1.ABS}.
 We show here that {\em every almost discrete space contains a maximal
discrete subspace and every such subspace is a retract of the enveloping
space}. Moreover, {\em if $X_{0}$ is a maximal discrete subspace of
an almost discrete space $X$ and $r : X \rightarrow X_{0}$ is
a continuous retraction, then $r^{-1}(x) = \overline{\{x\}}$ for every
point $x$ of $X$ belonging to $X_{0}$}.
This fact is a specialization, in the case of almost
discrete spaces, of the theorem of M.H. Stone that every topological
space can be made into a $T_{0}$-space by suitable identification
of points (see \cite{STONE:3}).}}


@ARTICLE{STONE:3,
       AUTHOR = {M. H. Stone},
        TITLE = {Application of Boolean algebras to topology},
      JOURNAL = {Math. Sb.},
         YEAR = {1936},
       VOLUME = {1},
        PAGES = {765--771}}
