@MISC{YELLOW19.MIZ,
  AUTHOR = {Bancerek, Grzegorz and Endou, Noboru and Sakai, Yuji},
  TITLE = {On the Characterizations of Compactness},
  DAY = {29},
  MONTH = {July},
  YEAR = {2001},
  ADDRESS1 = {University of Bia{\l}ystok},
  ADDRESS2 = {Gifu National College of Technology},
  ADDRESS3 = {Shinshu University\\Nagano},
  SUMMARY = {In the paper we show equivalence of the convergence of filters
 on a topological space and the convergence of nets in the space.
 We also give, five characterizations of compactness. Namely,
 for any topological space $T$ we proved that following condition
 are equivalent:
 \begin{itemize}
  \itemsep-3pt
 \item $T$ is compact,
 \item every ultrafilter on $T$ is convergent,
 \item every proper filter on $T$ has cluster point,
 \item every net in $T$ has cluster point,
 \item every net in $T$ has convergent subnet,
 \item every Cauchy net in $T$ is convergent.
 \end{itemize}}}

