On the Characterizations of Compactness
نویسندگان
چکیده
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: • T is compact, • every ultrafilter on T is convergent, • every proper filter on T has cluster point, • every net in T has cluster point, • every net in T has convergent subnet, • every Cauchy net in T is convergent.
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