@MISC{TDLAT_2.MIZ,
  AUTHOR = {Karno, Zbigniew and Watanabe, Toshihiko},
  TITLE = {Completeness of the Lattices of Domains 
of a Topological Space},
  NOTE = {This paper was done while the second author was visiting
 the Institute of Mathematics of Warsaw University in Bia{\l}ystok.},
  ACKNOWLEDGEMENT = {The authors would like to thank to Professors
  A.~Trybulec and Cz.~Byli\'nski for many helpful conversations during
  the preparation of this paper. The authors are also very grateful to
  G.~Bancerek for acquainting them with the {MIZ\TeX}\
  interface system for automated
  typesetting and translation.},
  SECTION1 = {Preliminary Theorems about Subsets of Topological Spaces},
  SECTION2 = {The Closure and the Interior Operations for Families\\ of Subsets of a Topological Space},
  SECTION3 = {Selected Properties of Domains of a Topological Space},
  SECTION4 = {Completeness of the Lattice of Domains},
  SECTION5 = {Completeness of the Lattices of Closed Domains \\ and Open Domains},
  DAY = {16},
  MONTH = {July},
  YEAR = {1992},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  ADDRESS2 = {Shinshu University\\Nagano},
  SUMMARY = {Let $T$ be a topological space and let $A$ be a subset of $T$.
Recall that $A$ is said to be a {\em domain} in $T$ provided
 ${\rm Int}\,\overline{A} \subseteq A \subseteq \overline{{\rm Int}\,A}$
(see \cite{TOPS_1.ABS} and comp. \cite{ISOMICHI}). This notion is 
a simple generalization of the
notions of open and closed domains in $T$ (see \cite{TOPS_1.ABS}).
Our main result is concerned with an extension of the following
well--known theorem
(see e.g. \cite{BIRKHOFF:1}, \cite{MOST-KURAT:3}, \cite{ENGEL:1}).
For a given topological space the Boolean lattices of all its closed
domains and all its open domains are complete.
It is proved here, using Mizar System, that {\em the complemented
lattice of all domains of a given topological space is complete}, too
(comp. \cite{TDLAT_1.ABS}).\par
It is known that both the lattice of open domains and the lattice of
closed domains are sublattices of the lattice of all domains
 \cite{TDLAT_1.ABS}. However, the following two problems remain open.
 \begin{itemize} \item[ ]
{\bf Problem 1.} Let $L$ be a sublattice of the lattice of all
domains. Suppose $L$ is complete, is smallest with respect to 
inclusion, and contains as sublattices the lattice 
of all closed domains and the lattice of all open domains.
Must $L$ be equal to the lattice of all domains~?
 \end{itemize}
A domain in $T$ is said to be a {\em Borel domain} provided
it is a Borel set. Of course every open (closed) domain is a Borel domain.
It can be proved that all Borel domains form a sublattice of the
lattice of domains.
 \begin{itemize} \item[ ]
{\bf Problem 2.} Let $L$ be a sublattice of the lattice of all
domains. Suppose $L$ is smallest with respect to inclusion and
contains as sublattices the lattice of all closed domains and the
lattice of all open domains.
Must $L$ be equal to the lattice of all Borel domains~?
 \end{itemize}
Note that in the beginning the closure and the interior operations for families
of subsets of topological spaces are introduced and their important
properties are presented (comp. \cite{KURAT:2}, \cite{KURAT:4},
 \cite{MOST-KURAT:3}). Using these notions, certain properties of domains,
closed domains and open domains are studied (comp. \cite{KURAT:4},
 \cite{ENGEL:1}).}}

