@MISC{TOPS_3.MIZ,
  AUTHOR = {Karno, Zbigniew},
  TITLE = {Remarks on Special Subsets of Topological Spaces},
  SECTION1 = {Selected Properties of Subsets of a Topological Space},
  SECTION2 = {Special Subsets of a Topological Space},
  SECTION3 = {Properties of Subsets in Subspaces},
  SECTION4 = {Subsets in Topological Spaces with the same Topological Structures},
  DAY = {06},
  MONTH = {April},
  YEAR = {1993},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  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 \cite{KURAT: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. \cite{TOPS_1.ABS}).
 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.}}

