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 i t i s not closed under direct product. I n t his paper, we gener-alize t he class of ( noncommutatively) s lender 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 π1 (X , x 0 ) is (weakly) noncommutatively slender, t hen X is semilocally simply connected (homotopically Hausdorff ) at x 0 .

Keywords