The Lattice of Domains of an Extremally Disconnected Space

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

The Lattice of Domains of an Extremally Disconnected Space

Zbigniew Karno: Warsaw University, Bialystok

Summary.

Let $X$ be a topological space and let $A$ be a subset of $X$. Recall that $A$ is said to be a {\em domain}\/ in $X$ provided ${\rm Int}\, \overline{A} \subseteq A \subseteq \overline{{\rm Int}\,A}$ (see [19], [7]). Recall also that $A$ is said to be a(n) {\em closed}\/ ({\em open})\/ {\em domain}\/ in $X$ if $A = \overline{{\rm Int}\,A}$ ($A = {\rm Int}\,\overline{A}$, resp.) (see e.g. [9], [19]). It is well-known that for a given topological space all its closed domains form a Boolean lattice, and similarly all its open domains form a Boolean lattice, too (see e.g., [10], [2]). In [17] it is proved that all domains of a given topological space form a complemented lattice. One may ask whether the lattice of all domains is Boolean. The aim is to give an answer to this question.\par To present the main results we first recall the definition of a class of topological spaces which is important here. $X$ is called {\em extremally disconnected}\/ if for every open subset $A$ of $X$ the closure $\overline {A}$ is open in $X$ [13] (comp. [6]). It is shown here, using Mizar System, that {\em the lattice of all domains of a topological space $X$ is modular iff $X$ is extremally disconnected.} Moreover, for every extremally disconnected space the lattice of all its domains coincides with both the lattice of all its closed domains and the lattice of all its open domains. From these facts it follows that {\em the lattice of all domains of a topological space $X$ is Boolean iff $X$ is extremally disconnected.}\par Note that we also review some of the standard facts on discrete, anti-discrete, almost discrete, extremally disconnected and hereditarily extremally disconnected topological spaces (comp. [9], [6]).

{\it Editor's Note:} This work has won the 1992 \'Sleszy\'nski's Award of the Mizar Society.

MML Identifier: TDLAT_3

The terminology and notation used in this paper have been introduced in the following articles [14] [5] [16] [11] [18] [12] [19] [3] [15] [4] [20] [1] [17] [8]

Contents (PDF format)

Selected Properties of Subsets of a Topological Space
Discrete Topological Structures
Discrete Topological Spaces
Extremally Disconnected Topological Spaces
The Lattice of Domains of Extremally Disconnected Spaces

Acknowledgments

The author wishes to thank to Professor A.~Trybulec for many helpful conversations during the preparation of this paper. The author is also very grateful to Cz.~Byli\'nski for acquainting him with new PC Mizar utilities programs.