Journal of Formalized Mathematics
Volume 5, 1993
University of Bialystok
Copyright (c) 1993 Association of Mizar Users

## Remarks on Special Subsets of Topological Spaces

Zbigniew Karno
Warsaw University, Bialystok

### Summary.

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.

#### MML Identifier: TOPS_3

The terminology and notation used in this paper have been introduced in the following articles [4] [6] [3] [7] [5] [1]

#### Contents (PDF format)

1. Selected Properties of Subsets of a Topological Space
2. Special Subsets of a Topological Space
3. Properties of Subsets in Subspaces
4. Subsets in Topological Spaces with the same Topological Structures

#### Bibliography

[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.