In this paper, we extend some results of D. Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power $2^{{\aleph }_0}$ commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.

We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by “nice” continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather...

We construct in Bell-Kunen’s model: (a) a group maximal topology on a countable infinite Boolean group of weight ${\aleph }_1<ℭ$ and (b) a countable irresolvable dense subspace of $2^{{\omega }_1}$. In this model $ℭ={\aleph }_{{\omega }_1}$.