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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.

General solutions of two functional inequalities and converses to contraction theorems

Jachymski J.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 2

147-156

We give necessary and sufficient conditions for the existence of some general solutions of the following two functional inequalities with an unknown real function phi: phi(Fx) is less than or equal to eta(phi(x)) and gamma(phi(Fx)) is less than or equal to phi(x). As an application we establish reciprocals to fixed point theorems of Matkowski and Wong. This extends an earlier result of Bessaga on a converse to the Banach contraction principle.

Remarks on a paper by Beg, latif and Minhas about fixed points

Jachymski J.

Demonstratio Mathematica

1998

Vol. 31, nr 2

333-338