Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

## Completeness of the Lattices of Domains of a Topological Space

Zbigniew Karno
Warsaw University, Bialystok
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 domain} in $T$ provided ${\rm Int}\,\overline{A} \subseteq A \subseteq \overline{{\rm Int}\,A}$ (see  and comp. ). This notion is a simple generalization of the notions of open and closed domains in $T$ (see ). Our main result is concerned with an extension of the following well-known theorem (see e.g. , , ). 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. ).\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 . 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. , , ). Using these notions, certain properties of domains, closed domains and open domains are studied (comp. , ).

This paper was done while the second author was visiting the Institute of Mathematics of Warsaw University in Bia{\l}ystok.

#### MML Identifier: TDLAT_2

The terminology and notation used in this paper have been introduced in the following articles             

#### Contents (PDF format)

1. Preliminary Theorems about Subsets of Topological Spaces
2. The Closure and the Interior Operations for Families\\ of Subsets of a Topological Space
3. Selected Properties of Domains of a Topological Space
4. Completeness of the Lattice of Domains
5. Completeness of the Lattices of Closed Domains \\ and Open Domains

#### Acknowledgments

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.

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