Extensions of Borel measurable maps and ranges of Borel bimeasurable maps

• Autorzy: Holicky P.
• Języki publikacji: EN

Abstrakty

EN

We prove an abstract version of the Kuratowski extension theorem for Borel measurable maps of a given class. It enables us to deduce and improve its nonseparable version due to Hansell. We also study the ranges of not necessarily injective Borel bimea-surable maps f and show that some control on the relative classes of preimages and images of Borel sets under f enables one to get a bound on the absolute class of the range of f. This seems to be of some interest even within separable spaces.

Słowa kluczowe

EN

absolute Borel sets; Borel classes; Borel isomorphisms; Borel bimeasurable maps; nonseparable spaces; scattered families

PL

zbiór borelowski absolutny; klasa Borela; izomorfizm Borela; przestrzeń nierozdzielona

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2004
• Tom: Vol. 52, nr 2
• Strony: 151--167
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Holicky P.

Bibliografia

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• [2] -, On Borel mappings and Baire functions, ibid. 194 (1974), 195-211.
• [3] -, On characterizing non-separable analytic and extended Borel sets as types of continuous images, Proc. London Math. Soc. 28 (1974), 683-699.
• [4] -, Descriptive topology, in: M. Hušek and J. van Mill (eds.), Recent Progress In General Topology, North-Holland, Amsterdam, 1992, 275-315.
• [5] -, Descriptive sets and the topology of nonseparable Banach spaces, Serdica Math. J. 27 (2001), 1-66.
• [6] P. Holický, Čech analytic and almost K-descriptive spaces, Czechoslovak Math. J. 43 (118) (1993), 451-466.
• [7] -, Borel maps with the „point of continuity property" and completely Borel additive families in some nonmetrizable spaces, Proc. Amer. Math. Soc. 120 (1994), 951-958.
• [8] P. Holický and J. Pelant, Internal descriptions of absolute Borel classes, to appear.
• [9] K. Kuratowski, Topology, Vol. I , Academic Press, New York, 1966.
• [10] W. Marciszewski and J. Pelant, Absolute Borel classes and function spaces, Trans. Amer. Math. Soc. 349 (1997), 3585-3596.
• [11] E. Michael, Some refinements of a selection theorem with 0-dimensional domain, Fund. Math. 140 (1992), 279-287.