@MISC{TEX_1.MIZ,
  AUTHOR = {Karno, Zbigniew},
  TITLE = {On Discrete and Almost Discrete Topological Spaces},
  ACKNOWLEDGEMENT = {The author wishes to thank to Professor
A. Trybulec for many helpful conversations during the preparation of
this paper.},
  SECTION1 = {Properties of Subsets of a Topological Space with Modified Topology},
  SECTION2 = {Trivial Topological Spaces},
  SECTION3 = {Examples of Discrete and Anti-discrete Topological Spaces},
  SECTION4 = {An Example of a Topological Space},
  SECTION5 = {Discrete and Almost Discrete Spaces},
  DAY = {06},
  MONTH = {April},
  YEAR = {1993},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  SUMMARY = {A topological space $X$ is called {\em almost discrete}\/ if every
open subset of $X$ is closed; equivalently, if every closed subset
of $X$ is open (comp. \cite{KURAT:2},\cite{KURAT:3}).
Almost discrete spaces were investigated in Mizar formalism in
 \cite{TDLAT_3.ABS}. We present here a few properties of such spaces
supplementary to those given in \cite{TDLAT_3.ABS}.\par
 Most interesting is the following characterization~: {\em A topological
space $X$ is almost discrete iff every nonempty subset of $X$ is
not nowhere dense}. Hence, {\em $X$ is non almost discrete iff there
is an everywhere dense subset of $X$ different from
the carrier of $X$}. We have an analogous characterization of
discrete spaces~: {\em A topological
space $X$ is discrete iff every nonempty subset of $X$ is
not boundary}. Hence, {\em $X$ is non discrete iff there
is a dense subset of $X$ different from the carrier of $X$}.
It is well known that the class of all almost discrete spaces contains
both the class of discrete spaces and the class of anti-discrete spaces
 (see e.g., \cite{TDLAT_3.ABS}). Observations presented here show that
the class of all almost discrete non discrete spaces
is not contained in the class of anti-discrete spaces and the class of
all almost discrete non anti-discrete spaces is not contained in the class
of discrete spaces. Moreover, the class of almost discrete non discrete
non anti-discrete spaces is nonempty. To analyse these
interdependencies we use various examples of topological spaces constructed
here in Mizar formalism.}}

