Isaac Namioka :
Fragmentability in Banach Spaces: Interactions of Topologies
Let (X,\tau) be a topological space and let \rho be a metric
on X. In their 1985 paper, Jayne and Rogers define (X,\tau) to be
fragmented by \rho if each non-empty subset of X admits non-empty
relatively \tau-open subsets of arbitrarily small \rho-diameter.
Prior to 1985, many mathematicians had encountered instances of
fragmentability in connection with Ryll-Nardzewski's fixed point
theorem, Banach spaces with Radon-Nikodym property and Asplund
spaces, but they were without the convenient term "fragmented".
I shall begin my lectures by reviewing the pre-1985 topics
mentioned above from the vantage point of fragmentability. Then I
will talk on examples and applications of fragmentability in Banach
spaces. Then I will introduce the notion of sigma-fragmentabiliy and
will discuss its uses and examples in connection with Banach and
function spaces. I shall discuss a recent work on renorming by Raja
that uses a variant of sigma-fragmentability. Finally I will talk on
curious connections between the Lindelof property and
fragmentability.