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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE XLV (1) (2005), 33-41 A covering criterion and its applications to product theorems
unpublished
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We establish a covering criterion involving a neighbourhood system and ideals of open sets which yields, in particular, a compactness criterion for an arbitrary topological space. As an application, we give new proofs of Tychonoff's compactness theorem: we consider separately the case of a countable product, in a proof of which the ordinary mathematical induction is used, and the case of an uncountable product proved by the transfinite induction. Subsequently, the same argument is applied to
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