On a problem of Talagrand concerning separately continuous functions

Speaker(s)

Roman Pol

Affiliation

University of Warsaw

Date

Oct. 17, 2018, 4:15 p.m.

Room

room 5050

Seminar

Topology and Set Theory Seminar

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This is a joint work with Volodymyr Mykhaylyuk (a preliminary version of our paper is on arXiv: https://arxiv.org/pdf/1809.05799.pdf)
  M.Talagrand [Math.Ann.270(1985),Probleme 3] stated the following problem (recalled recently in Guirao, V.Montesinos, V.Zizler, Open problems in geometry and analysis of Banach spaces, Springer (2016),Problem 285) :
  Let f : E x K ---> R be a separately continuous function on the product of a Baire space E and a compact space K. Does f have a point of joint continuity ?

   A negative solution was obtained by V. Mykhaylyuk and myself several weeks ago. In fact, using an approach of Mykhaylyuk from [Mat.Studii (2008)] and a construction of Kunen, van Mill and Mills [PAMS (1980)] we proved the following stronger result.

    Theorem. There exists s separately continuous function e : E x K ---> {0,1} on the product of a Baire space E and a zero-dimensional compact space K such that no restriction of e to any non-meager Borel set in E x K is continuous.

  In particular, the function e fails the Baire property, and hence has no points of joint continuity on some nonempty open-and closed rectangle in E x K.
   We shall present a proof of this theorem and some comments on this topic.

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