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A covering criterion and its applications to product theorems

Jachymski J.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2005

Vol. 45, [Z] 1

35--43

We establish a covering criterion involving a neighbourhood system and idealsof open sets which yields,in particular, a compactness criterion for an arbitrary topologicalspace. As an application, we give new proofsof Tychonoff's compactness theorem: we consider separately the case of acountable product, in a proof of which the ordinary mathematical induction isused, and the case of an uncountable product proved by the transfiniteinduction. Subsequently, the same argument is applied to obtain some resultson products of Lindelöf spaces.