Namioka

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.