@MISC{TDLAT_1.MIZ,
  AUTHOR = {Watanabe, Toshihiko},
  TITLE = {The Lattice of Domains of a Topological Space},
  NOTE = {This paper was done under the supervision of Z. Karno while the
 author was visiting the Institute of Mathematics of Warsaw
 University in Bia{\l}ystok.},
  ACKNOWLEDGEMENT = {The author wishes to express his thanks to Professors
  A.~Trybulec and Z.~Karno for their active interest in the publication
  of this article and for useful suggestions and many valuable comments.},
  SECTION1 = {Preliminary Theorems on Subset of Topological Spaces},
  SECTION2 = {Properties of Domains of Topological Spaces},
  SECTION3 = {The Lattice of Domains},
  SECTION4 = {The Lattice of Closed Domains},
  SECTION5 = {The Lattice of Open Domains},
  SECTION6 = {Connections between Lattices of Domains},
  DAY = {12},
  MONTH = {June},
  YEAR = {1992},
  ADDRESS1 = {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 closed domain} of $T$ if
 $A = \overline{{\rm Int}\,A}$  and
 $A$ is said to be an {\em open domain} of $T$ if
 $A = {\rm Int}\,\overline{A}$
(see e.g. \cite{KURAT:2}, \cite{TOPS_1.ABS}).
Some simple generalization of these notions is the following one.
 $A$ is said to be a {\em domain} of $T$ provided
 ${\rm Int}\,\overline{A} \subseteq A \subseteq \overline{{\rm Int}\,A}$
(see \cite{TOPS_1.ABS} and compare \cite{ISOMICHI}).
In this paper certain connections between these concepts are introduced
and studied. \par
Our main results are concerned with the following
well--known theorems (see e.g. \cite{MOST-KURAT:3}, \cite{BIRKHOFF:1}).
For a given topological space all its closed domains form a Boolean
lattice, and similarly all its open domains form a Boolean lattice, too.
It is proved that {\em all domains of a given topological space form
a complemented lattice.}
Moreover, it is shown that both {\em the lattice of open domains and the
lattice of closed domains are sublattices of the lattice of all domains.}
In the beginning some useful theorems about subsets of topological spaces
are proved and certain properties of domains, closed domains and open
domains are discussed.}}

