The Lattice of Domains of a Topological Space
Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992
Association of Mizar Users
The Lattice of Domains of a Topological Space
-
Toshihiko Watanabe
-
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. [8], [14]).
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 [14] and compare [7]).
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. [9], [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.
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.
MML Identifier:
TDLAT_1
The terminology and notation used in this paper have been
introduced in the following articles
[11]
[5]
[12]
[10]
[15]
[2]
[14]
[13]
[3]
[4]
[6]
-
Preliminary Theorems on Subset of Topological Spaces
-
Properties of Domains of Topological Spaces
-
The Lattice of Domains
-
The Lattice of Closed Domains
-
The Lattice of Open Domains
-
Connections between Lattices of Domains
Acknowledgments
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.
Bibliography
- [1]
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Received June 12, 1992
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