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

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