On weak-fragmentability of Banach spaces (Q2925691)

The authors discuss some properties of Banach spaces related to fragmentability. They prove that (i) each countable \(T_1\)-space \(X\) is fragmentable by a metric that generates the discrete topology on \(X\), and (ii) a hereditarily Baire topological space \(X\) is scattered if and only if it is fragmentable by a metric that generates the discrete topology on \(X\). Applying these results to Banach spaces, it is proved that the weak topology of any non-trivial normed linear space is not fragmentable by a metric generating the discrete topology. Finally, looking at the Banach space \(X=l_\infty/c_0\), the authors observe that the fragmentability of the weak topology of the dual Banach space \(X^*\) does not imply the (sigma-)fragmentability of \(X\).