We disprove the existence of a universal object in several classes of spaces including the class of weakly Lindelöf Banach spaces.

In the previous paper, we, together with J. Orihuela, showed that a compact subset X of the product space ${[-1,1]}^D$ is fragmented by the uniform metric if and only if X is Lindelöf with respect to the topology γ(D) of uniform convergence on countable subsets of D. In the present paper we generalize the previous result to the case where X is K-analytic. Stated more precisely, a K-analytic subspace X of ${[-1,1]}^D$ is σ-fragmented by the uniform metric if and only if (X,γ(D)) is Lindelöf, and if this is the case then...

A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^D$ the following four conditions are equivalent: (i) K is fragmented by $d_D$, where, for each S ⊂ D, $d_S(x,y)=sup\varrho \left(x\right(t),y(t\left)\right):t\in S$. (ii) For each countable subset A of D, $(K,d_A)$ is...