On Weakly (Noncommutatively) Slender Groups

Abstract

The class of slender groups is closed under direct sums, but it is not closed under direct product. Also, the class of noncommutatively slender groups is closed under weak direct products, but it is not closed under direct product. In this paper, we generalize the class of (noncommutatively) slender groups to weakly (noncommutatively) slender groups, which is closed under direct products and inverse limits. Also, we show that for a topological space X with first countability at $$x_0$$ , if $$\pi _1 (X,x_0)$$ is (weakly) noncommutatively slender, then X is semilocally simply connected (homotopically Hausdorff) at $$x_0$$ .