@MISC{TEX_4.MIZ,
  AUTHOR = {Karno, Zbigniew},
  TITLE = {Maximal Anti-Discrete Subspaces of Topological Spaces},
  SECTION1 = {Properties of the Closure and the Interior Operations},
  SECTION2 = {Anti-Discrete Subsets of Topological Structures},
  SECTION3 = {Maximal Anti-Discrete Subsets of Topological Structures},
  SECTION4 = {Anti-Discrete and Maximal Anti-discrete Subsets of Topological Spaces},
  SECTION5 = {Maximal Anti-Discrete Subspaces},
  DAY = {26},
  MONTH = {July},
  YEAR = {1994},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  SUMMARY = {Let $X$ be a topological space and let $A$ be a subset of $X$.
 $A$ is said to be {\em anti-discrete}\/ provided for every open subset
 $G$ of $X$ either $A \cap G = \emptyset$ or $A \subseteq G$;
equivalently, for every closed subset $F$ of $X$ either
 $A \cap F = \emptyset$ or $A \subseteq F$.
 An anti-discrete subset $M$ of $X$ is said to be
{\em maximal anti-discrete}\/ provided for every anti-discrete
subset $A$ of $X$ if $M \subseteq A$ then $M = A$.
A subspace of $X$ is {\em maximal anti-discrete}\/
iff its carrier is maximal anti-discrete in $X$.
 The purpose is to list a few properties of maximal anti-discrete sets
and subspaces in Mizar formalism.\par
 It is shown that every $x \in X$ is contained in a unique maximal
anti-discrete subset M$(x)$ of $X$, denoted in the text by MaxADSet($x$).
Such subset can be defined by
 $${\rm M}(x) = \bigcap\ \{ S \subseteq X : x \in S,\ {\rm and}\ S
    \ {\rm is}\ {\rm open}\ {\rm or}\ {\rm closed}\  {\rm in}\ X\}.$$
 It has the following remarkable properties:
  (1) $y \in {\rm M}(x)$ iff ${\rm M}(y) = {\rm M}(x)$,
  (2) either ${\rm M}(x) \cap {\rm M}(y) = \emptyset$ or
      ${\rm M}(x) = {\rm M}(y)$,
  (3) ${\rm M}(x) = {\rm M}(y)$ iff
        $\overline{\{x\}} = \overline{\{y\}}$, and
  (4) ${\rm M}(x) \cap {\rm M}(y) = \emptyset$ iff
        $\overline{\{x\}} \neq \overline{\{y\}}$.
 It follows from these properties that
  $\{{\rm M}(x) : x \in X\}$
 is the $T_{0}$-partition of $X$ defined by M.H.~Stone in \cite{STONE:3}.\par
 Moreover, it is shown that the operation M defined on all subsets
of $X$ by
 $${\rm M}(A) = \bigcup\ \{{\rm M}(x) : x \in A\},$$
denoted in the text by MaxADSet($A$),
satisfies the Kuratowski closure axioms (see e.g., \cite{KURAT:2}), i.e.,
 (1) ${\rm M}(A \cup B) = {\rm M}(A) \cup {\rm M}(B)$,
 (2) ${\rm M}(A) = {\rm M}({\rm M}(A))$,
 (3) $A \subseteq {\rm M}(A)$,and
 (4) ${\rm M}(\emptyset) = \emptyset$.
 Note that this operation commutes with the usual closure
operation of $X$, and if $A$ is an open (or a closed) subset of $X$,
then ${\rm M}(A) = A$.}}


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