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.

Received April 6, 1993


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