Volume 13, 2001

University of Bialystok

Copyright (c) 2001 Association of Mizar Users

**Grzegorz Bancerek**- University of Bialystok
**Noboru Endou**- Gifu National College of Technology
**Yuji Sakai**- Shinshu University, Nagano

- 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}

Contents (PDF format)

- [1]
Grzegorz Bancerek.
Complete lattices.
*Journal of Formalized Mathematics*, 4, 1992. - [2]
Grzegorz Bancerek.
Bounds in posets and relational substructures.
*Journal of Formalized Mathematics*, 8, 1996. - [3]
Grzegorz Bancerek.
Directed sets, nets, ideals, filters, and maps.
*Journal of Formalized Mathematics*, 8, 1996. - [4]
Grzegorz Bancerek.
Prime ideals and filters.
*Journal of Formalized Mathematics*, 8, 1996. - [5]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Agata Darmochwal.
Compact spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Agata Darmochwal.
Families of subsets, subspaces and mappings in topological spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Adam Grabowski and Robert Milewski.
Boolean posets, posets under inclusion and products of relational structures.
*Journal of Formalized Mathematics*, 8, 1996. - [12]
Zbigniew Karno.
Maximal discrete subspaces of almost discrete topological spaces.
*Journal of Formalized Mathematics*, 5, 1993. - [13]
Artur Kornilowicz.
On the topological properties of meet-continuous lattices.
*Journal of Formalized Mathematics*, 8, 1996. - [14]
Beata Padlewska.
Families of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Beata Padlewska.
Locally connected spaces.
*Journal of Formalized Mathematics*, 2, 1990. - [16]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Alexander Yu. Shibakov and Andrzej Trybulec.
The Cantor set.
*Journal of Formalized Mathematics*, 7, 1995. - [18]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [19]
Andrzej Trybulec.
Tuples, projections and Cartesian products.
*Journal of Formalized Mathematics*, 1, 1989. - [20]
Andrzej Trybulec.
Moore-Smith convergence.
*Journal of Formalized Mathematics*, 8, 1996. - [21]
Wojciech A. Trybulec.
Partially ordered sets.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [23]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [24]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989. - [25]
Edmund Woronowicz and Anna Zalewska.
Properties of binary relations.
*Journal of Formalized Mathematics*, 1, 1989.

[ Download a postscript version, MML identifier index, Mizar home page]