Journal of Formalized Mathematics
Volume 13, 2001
University of Bialystok
Copyright (c) 2001
Association of Mizar Users
On the Characterizations of Compactness
-
Grzegorz Bancerek
-
University of Bialystok
-
Noboru Endou
-
Gifu National College of Technology
-
Yuji Sakai
-
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}
The terminology and notation used in this paper have been
introduced in the following articles
[18]
[7]
[22]
[23]
[19]
[14]
[10]
[5]
[25]
[24]
[6]
[16]
[9]
[12]
[8]
[15]
[17]
[21]
[1]
[2]
[3]
[11]
[4]
[20]
[13]
Contents (PDF format)
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Received July 29, 2001
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