@MISC{TOPGEN_1.MIZ,
  AUTHOR = {Grabowski, Adam},
  TITLE = {On the Boundary and Derivative of a Set},
  NOTE = {This work has been partially supported by the KBN grant
  4 T11C 039 24 and the FP6 IST grant TYPES No. 510996.},
  SECTION1 = {Preliminaries},
  SECTION2 = {Boundary of a Subset},
  SECTION3 = {Accumulation Points and Derivative of a Set},
  SECTION4 = {Isolated Points},
  SECTION5 = {Derivative of a Subset-Family},
  SECTION6 = {Density-in-itself},
  SECTION7 = {Perfect Sets},
  SECTION8 = {Scattered Subsets},
  SECTION9 = {Density of a Topological Space and Separable Spaces},
  SECTION10 = {Exercises},
  DAY = {08},
  MONTH = {November},
  YEAR = {2004},
  ADDRESS1 = {University of Bia{\l}ystok},
  SUMMARY = {This is the first Mizar article in a series
  aiming at a complete formalization of the textbook
  ``General Topology''
  by Engelking \cite{ENGEL:1}.
  We cover the first part of Section 1.3, by defining
  such notions as a derivative of a subset $A$ of a topological
  space (usually denoted by $A^{\rm d}$, but ${\rm Der~} A$ in our notation),
  the derivative and the
  boundary of families of subsets, points of accumulation and isolated
  points. We also introduce dense-in-itself, perfect and scattered
  topological spaces and formulate the notion of the density of a space.
  Some basic properties are given as well as selected exercises
  from \cite{ENGEL:1}.}}

