Marek Cúth

Separable reduction theorems by the method of elementary submodels

We introduce an interesting method of proving separable reduction theorems - the method of elementary submodels. We are studying whether it is true that a set (function) has given property if and only if it has this property with respect to a special separable subspace, dependent only on the given set (function). We are interested in properties of sets to be dense, nowhere dense, meager, residual or porous and in properties of functions to be continuous, semicontinuous or Frechet differentiable. Our method of creating separable subspaces enables us to combine our results, so we easily get separable reductions of function properties such as be continuous on a dense subset, be Frechet differentiable on a residual subset, etc. Finally, we show some applications of presented separable reduction theorems and demonstrate that some results of Zajicek, Lindenstrauss and Preiss hold in nonseparable setting as well.

On the structure of Lipschitz-free spaces

In this note we study the structure of Lipschitz-free Banach spaces. We show that every Lipschitz-free Banach space over an infinite metric space contains a complemented copy of

Simultaneous projectional skeletons

ell_1

Lipschitz-Free Spaces Over Ultrametric Spaces

. This result has many consequences for the structure of Lipschitz-free Banach spaces. Moreover, we give an example of a countable compact metric space

Monotone retractability and retractional skeletons

K

Rich families and elementary submodels

such that

On separable determination of σ-P-porous sets in Banach spaces

F(K)

Asplund spaces characterized by rich families and separable reduction of Fréchet subdifferentiability

is not isomorphic to a subspace of

Projections in duals to Asplund spaces made without Simons’ lemma

L_1

σ-Porosity is separably determined

and we show that whenever