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}

MML Identifier: YELLOW19

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)

Bibliography

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Received July 29, 2001


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