Topology, isomorphic smoothness and polyhedrality in Banach spaces

In recent decades, topology has come to play an increasing role in some geometric aspects of Banach space theory. The class of so-called $w^*$-locally relatively compact sets was introduced recently by Fonf, Pallares, Troyanski and the author, and were found to be a useful topological tool in the theory of isomorphic smoothness and polyhedrality in Banach spaces. We develop the topological theory of these sets and present some Banach space applications.