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2 Bulletin of the Polish Academy of Sciences. Mathematics

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A note on differentiability of Lipschitz maps

Górak R.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

Vol. 58, no 3

259-268

We show that every Lipschitz map defined on an open subset of the Banach space C(K), where K is a scattered compactum, with values in a Banach space with the Radon-Nikodym property, has a point of Frechet differentiability. This is a strengthening of the result of Lindenstrauss and Preiss who proved that for countable compacta. As a consequence of the above and a result of Arvanitakis we prove that Lipschitz functions on certain function spaces are Gateaux differentiable.

Countable compact scattered T2 spaces and weak forms of AC

Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

2006

Vol. 54, no 1

75-84

We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T2 topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T2 space is scattered iff it is metrizable. (3) If the real line R can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T2 space is scattered. (4) It is not provable in ZF+¬AC that there exists a countable compact T2 space which is dense-in-itself.