@MISC{TSEP_2.MIZ,
  AUTHOR = {Karno, Zbigniew},
  TITLE = {On a Duality Between Weakly Separated 
Subspaces of Topological Spaces},
  SECTION1 = {Certain Set--Decompositions of a Topological Space},
  SECTION2 = {Duality between Pairs of Weakly Separated Subsets},
  SECTION3 = {Certain Subspace--Decompositions of a Topological Space},
  SECTION4 = {Duality between Pairs of Weakly Separated Subspaces},
  DAY = {09},
  MONTH = {November},
  YEAR = {1992},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  SUMMARY = {Let $X$ be a topological space and let $X_{1}$ and
  $X_{2}$ be subspaces of $X$ with the carriers $A_{1}$ and
  $A_{2}$, respectively. Recall that $X_{1}$ and $X_{2}$
 are {\em weakly separated}\/ if $A_{1} \setminus A_{2}$ and
  $A_{2} \setminus A_{1}$ are separated (see \cite{TSEP_1.ABS}
 and also \cite{TMAP_1.ABS} for applications).
 Our purpose is to list a number of properties of such subspaces,
 supplementary to those given in \cite{TSEP_1.ABS}.
 Note that in the Mizar formalism the carrier of any topological
 space (hence the carrier of any its subspace) is always non--empty,
 therefore for convenience we list beforehand analogous properties
 of weakly separated subsets without any additional conditions.\par
 To present the main results we first formulate a useful definition.
 We say that $X_{1}$ and $X_{2}$ {\em constitute a decomposition}\/
 of $X$ if $A_{1}$ and $A_{2}$ are disjoint and the union of $A_{1}$
 and $A_{2}$ covers the carrier of $X$ (comp. \cite{KURAT:2}).
 We are ready now to present the following duality property between pairs
 of weakly separated subspaces~: {\em If each pair of subspaces
  $X_{1}$, $Y_{1}$ and $X_{2}$, $Y_{2}$ of $X$ constitutes a decomposition
 of $X$, then $X_{1}$ and $X_{2}$ are weakly separated iff $Y_{1}$ and
  $Y_{2}$ are weakly separated}. From this theorem we get
 immediately that under
 the same hypothesis, {\em $X_{1}$ and $X_{2}$ are separated iff
  $X_{1}$ misses $X_{2}$ and $Y_{1}$ and $Y_{2}$ are weakly separated}.
 Moreover, we show the following enlargement theorem~: {\em If
  $X_{i}$ and $Y_{i}$ are subspaces of $X$ such that $Y_{i}$ is a subspace
 of $X_{i}$ and $Y_{1} \cup Y_{2} = X_{1} \cup X_{2}$
 and if $Y_{1}$ and $Y_{2}$ are weakly separated, then $X_{1}$
 and $X_{2}$ are weakly separated}.
 We show also the following dual extenuation theorem~: {\em If
  $X_{i}$ and $Y_{i}$ are subspaces of $X$ such that $Y_{i}$ is a subspace
 of $X_{i}$ and $Y_{1} \cap Y_{2} = X_{1} \cap X_{2}$
 and if $X_{1}$ and $X_{2}$ are weakly separated, then $Y_{1}$
 and $Y_{2}$ are weakly separated}.
 At the end we give a few properties of weakly separated
 subspaces in subspaces.}}

