A topological space $(T,\tau)$ 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
$(M,\varrho)$ be a metric space with $\varrho$ 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 $S\subset D$,
$$d_{S}(x,y)=\sup\{\varrho(x(t),y(t)):
t\in S\}.$$(ii) For each countable subset $A$ of $D$,
$(K,d_{A})$ is separable.(iii) The space $(K,\gamma (D))$ is
Lindelöf, where $\gamma (D)$ is the topology of uniform
convergence on the family of countable subsets of $D$.(iv) $(K,\gamma (D))^{{\mathbb N}}$ is Lindelöf.The rest of the paper is devoted to applications of the basic
theorem. Here are some of them. A compact Hausdorff space $K$ is
Radon–Nikodým compact if, and only if, there is a bounded subset
$D$ of $C(K)$ separating the points of $K$ such that $(K,\gamma
(D))$ is Lindelöf. If $X$ is a Banach space and $H$ is a
weak$^{\ast}$-compact subset of the dual $X^\ast$ which is weakly
Lindelöf, then $(H,\hbox{weak})^{\mathbb N}$ is Lindelöf. Furthermore,
under the same condition $\overline{{\rm span}(H)}^{\,\|\ \|}$ and
$\overline{{\rm co}{(H)}}^{\,w^{\ast}}$ are weakly Lindelöf. The last
conclusion answers a question by Talagrand. Finally we apply our
basic theorem to certain classes of Banach spaces including weakly
compactly generated ones and the duals of Asplund spaces.
