A covering criterion and its applications to product theorems

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 obtain some results on products of Lindelöf spaces.