@MISC{ROUGHS_4.MIZ,
  AUTHOR = {Grabowski, Adam},
  TITLE = {Topological Interpretation of Rough Sets},
  SECTION1 = {Preliminaries},
  SECTION2 = {Ordinary Properties of Maps},
  SECTION3 = {Structural Part},
  SECTION4 = {Merging with Topologies},
  SECTION5 = {Introducing Rough Sets},
  SECTION6 = {On Sequential Closure and Frechet Spaces},
  SECTION7 = {Connections between Closures and Approximations},
  SECTION8 = {Isomichi Results Reuse},
  SECTION9 = {Reexamination of Kuratowski's 14 Sets for Approximation Spaces},
  DAY = {31},
  MONTH = {March},
  YEAR = {2014},
  ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok\\Poland},
  SUMMARY = {Rough sets, developed by Pawlak, are important model of
  incomplete or partially known information.
  In this article, which is essentially a continuation of
  \cite{ROUGHS_1.ABS}, we characterize rough sets in
  terms of topological closure and interior, as the approximations
  have the properties of these Kuratowski operators.
  We decided to merge topological spaces with tolerance approximation
  spaces. As a testbed for our developed approach we restated
  results of Isomichi \cite{ISOMICHI} (formalized in Mizar in
  \cite{ISOMICHI}) and
  about fourteen sets of Kuratowski \cite{KURAT:4}
  (encoded with the help of
  Mizar adjectives and clusters' registrations in \cite{KURATO_1.ABS})
  in terms of rough approximations. The upper bounds which were
  14 and 7 in the original paper of Kuratowski, in our case are six
  and three, respectively.
  \par
  It turns out that within the classification given by Isomichi,
  $1^{\rm st}$ class subsets are precisely crisp sets,
  $2^{\rm nd}$ class subsets are proper rough sets, and
  there are no $3^{\rm rd}$ class subsets in topological spaces
  generated by approximations. Also the important results about these
  spaces is that they are extremally disconnected \cite{TDLAT_3.ABS},
  hence lattices of their domains are Boolean.}}

