Journal of Formalized Mathematics
Volume 5, 1993
University of Bialystok
Copyright (c) 1993
Association of Mizar Users
Remarks on Special Subsets of Topological Spaces
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Zbigniew Karno
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Warsaw University, Bialystok
Summary.
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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]
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Selected Properties of Subsets of a Topological Space
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Special Subsets of a Topological Space
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Properties of Subsets in Subspaces
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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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