Volume 5, 1993

University of Bialystok

Copyright (c) 1993 Association of Mizar Users

**Zbigniew Karno**- Warsaw University, Bialystok

- Let $X$ be a topological space and let $A$ be a subset of $X$. Recall that $A$ is {\em nowhere dense}\/ in $X$ if its closure is a boundary subset of $X$, i.e., if ${\rm Int}\,\overline{A} = \emptyset$ (see [2]). We introduce here the concept of everywhere dense subsets in $X$, which is dual to the above one. Namely, $A$ is said to be {\em everywhere dense}\/ in $X$ if its interior is a dense subset of $X$, i.e., if $\overline{{\rm Int}\,A} =$ the carrier of $X$.\par Our purpose is to list a number of properties of such sets (comp. [7]). As a sample we formulate their two dual characterizations. The first one characterizes thin sets in $X$~: {\em $A$ is nowhere dense iff for every open nonempty subset $G$ of $X$ there is an open nonempty subset of $X$ contained in $G$ and disjoint from $A$}. The corresponding second one characterizes thick sets in $X$~: {\em $A$ is everywhere dense iff for every closed subset $F$ of $X$ distinct from the carrier of $X$ there is a closed subset of $X$ distinct from the carrier of $X$, which contains $F$ and together with $A$ covers the carrier of $X$}. We also give some connections between both these concepts. Of course, {\em $A$ is everywhere (nowhere) dense in $X$ iff its complement is nowhere (everywhere) dense}. Moreover, {\em $A$ is nowhere dense iff there are two subsets of $X$, $C$ boundary closed and $B$ everywhere dense, such that $A = C \cap B$ and $C \cup B$ covers the carrier of $X$}. Dually, {\em $A$ is everywhere dense iff there are two disjoint subsets of $X$, $C$ open dense and $B$ nowhere dense, such that $A = C \cup B$}.\par Note that some relationships between everywhere (nowhere) dense sets in $X$ and everywhere (nowhere) dense sets in subspaces of $X$ are also indicated.

- Selected Properties of Subsets of a Topological Space
- Special Subsets of a Topological Space
- Properties of Subsets in Subspaces
- Subsets in Topological Spaces with the same Topological Structures

- [1]
Zbigniew Karno.
Separated and weakly separated subspaces of topological spaces.
*Journal of Formalized Mathematics*, 4, 1992. - [2] Kazimierz Kuratowski. \em Topology, volume I. PWN - Polish Scientific Publishers, Academic Press, Warsaw, New York and London, 1966.
- [3]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [5]
Andrzej Trybulec.
A Borsuk theorem on homotopy types.
*Journal of Formalized Mathematics*, 3, 1991. - [6]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Miroslaw Wysocki and Agata Darmochwal.
Subsets of topological spaces.
*Journal of Formalized Mathematics*, 1, 1989.

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