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

Contents (PDF format)

1. Preliminary Theorems on Subset of Topological Spaces
2. Properties of Domains of Topological Spaces
3. The Lattice of Domains
4. The Lattice of Closed Domains
5. The Lattice of Open Domains
6. Connections between Lattices of Domains

Acknowledgments

Bibliography

[1] Garrett Birkhoff. \em Lattice Theory. Providence, Rhode Island, New York, 1967.

[2] Czeslaw Bylinski. Binary operations. Journal of Formalized Mathematics, 1, 1989.

[3] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[4] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.

[5] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.

[6] Marek Chmur. The lattice of natural numbers and the sublattice of it. The set of prime numbers. Journal of Formalized Mathematics, 3, 1991.

[7] Yoshinori Isomichi. New concepts in the theory of topological space -- supercondensed set, subcondensed set, and condensed set. \em Pacific Journal of Mathematics, 38(3):657--668, 1971.

[8] Kazimierz Kuratowski. \em Topology, volume I. PWN - Polish Scientific Publishers, Academic Press, Warsaw, New York and London, 1966.

[9] Kazimierz Kuratowski and Andrzej Mostowski. \em Set Theory (with an introduction to descriptive set theory), volume 86 of \em Studies in Logic and The Foundations of Mathematics. PWN - Polish Scientific Publishers and North-Holland Publishing Company, Warsaw-Amsterdam, 1976.

[10] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.

[11] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[12] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[13] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[14] Miroslaw Wysocki and Agata Darmochwal. Subsets of topological spaces. Journal of Formalized Mathematics, 1, 1989.

[15] Stanislaw Zukowski. Introduction to lattice theory. Journal of Formalized Mathematics, 1, 1989.