We discuss some recent results related to the basis problem and the extension problem for spaces of infinitely differentiable functions and their traces. In particular, we show that there is no characterization of the extension property (the existence of a continuous linear extension operator for Whitney spaces) in terms of densities of Hausdorff  contents or related characteristics.

Also we show that the space of Whitney jets defined on a compact set does not possess a polynomial basis in the case of sets of infinite cardinality with isolated points. Thus, in general, there is no stability of bases in NF-spaces.

The research is partially supported by TÜBİTAK, Project 119F023.