We show that in all infinite-dimensional normed spaces it is possible to construct a fixed point free continuous map of the unit ball whose measure of noncompactness is bounded by 2. Moreover, for a large class of spaces (containing separable spaces, Hilbert spaces and l-infinity (S)) even the best possible bound 1 is attained for certain measures of noncompactness.

Let K be a compact Hausdorff space, $C_p\left(K\right)$ the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and $t_p\left(D\right)$ the topology in C(K) of pointwise convergence on D. It is proved that when $C_p\left(K\right)$ is Lindelöf the $t_p\left(D\right)$-compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and $C_p\left(K\right)$ is Lindelöf, then K is metrizable if, and only if, there is a countable and dense...

Recent work has studied the fragmentability and σ-fragmentability properties of Banach spaces. Here examples are given that justify the definitions that have been used. The fragmentability and σ-fragmentability properties of the spaces ${\ell }^{\infty }$ and ${\ell }_c^{\infty }\left(\Gamma \right)$, with Γ uncountable, are determined.