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)

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