Descriptive properties of mappings between nonseparable Luzin spaces

Petr Holický; Václav Komínek

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 1, page 201-224
  • ISSN: 0011-4642

Abstract

top
We relate some subsets G of the product X × Y of nonseparable Luzin (e.g., completely metrizable) spaces to subsets H of ℕ ℕ × Y in a way which allows to deduce descriptive properties of G from corresponding theorems on H . As consequences we prove a nonseparable version of Kondô’s uniformization theorem and results on sets of points y in Y with particular properties of fibres f - 1 ( y ) of a mapping f X → Y . Using these, we get descriptions of bimeasurable mappings between nonseparable Luzin spaces in terms of fibres.

How to cite

top

Holický, Petr, and Komínek, Václav. "Descriptive properties of mappings between nonseparable Luzin spaces." Czechoslovak Mathematical Journal 57.1 (2007): 201-224. <http://eudml.org/doc/31125>.

@article{Holický2007,
abstract = {We relate some subsets $G$ of the product $X\times Y$ of nonseparable Luzin (e.g., completely metrizable) spaces to subsets $H$ of $\mathbb \{N\}^\{\mathbb \{N\}\}\times Y$ in a way which allows to deduce descriptive properties of $G$ from corresponding theorems on $H$. As consequences we prove a nonseparable version of Kondô’s uniformization theorem and results on sets of points $y$ in $Y$ with particular properties of fibres $f^\{-1\}(y)$ of a mapping $f\: X\rightarrow Y$. Using these, we get descriptions of bimeasurable mappings between nonseparable Luzin spaces in terms of fibres.},
author = {Holický, Petr, Komínek, Václav},
journal = {Czechoslovak Mathematical Journal},
keywords = {nonseparable metric spaces; Luzin spaces; $\sigma $-discrete network; uniformization; bimeasurable maps; nonseparable metric spaces; Luzin spaces; -discrete network; uniformization; bimeasurable maps},
language = {eng},
number = {1},
pages = {201-224},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Descriptive properties of mappings between nonseparable Luzin spaces},
url = {http://eudml.org/doc/31125},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Holický, Petr
AU - Komínek, Václav
TI - Descriptive properties of mappings between nonseparable Luzin spaces
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 201
EP - 224
AB - We relate some subsets $G$ of the product $X\times Y$ of nonseparable Luzin (e.g., completely metrizable) spaces to subsets $H$ of $\mathbb {N}^{\mathbb {N}}\times Y$ in a way which allows to deduce descriptive properties of $G$ from corresponding theorems on $H$. As consequences we prove a nonseparable version of Kondô’s uniformization theorem and results on sets of points $y$ in $Y$ with particular properties of fibres $f^{-1}(y)$ of a mapping $f\: X\rightarrow Y$. Using these, we get descriptions of bimeasurable mappings between nonseparable Luzin spaces in terms of fibres.
LA - eng
KW - nonseparable metric spaces; Luzin spaces; $\sigma $-discrete network; uniformization; bimeasurable maps; nonseparable metric spaces; Luzin spaces; -discrete network; uniformization; bimeasurable maps
UR - http://eudml.org/doc/31125
ER -

References

top
  1. 10.1090/S0002-9904-1970-12584-8, Bull. Amer. Math. Soc. 76 (1970), 1112–1117. (1970) MR0265539DOI10.1090/S0002-9904-1970-12584-8
  2. 10.4064/fm-117-3-165-185, Fund. Math. 117 (1983), 165–185. (1983) MR0719837DOI10.4064/fm-117-3-165-185
  3. On characterizing non-separable analytic and extended Borel sets as types of continuous images, Proc. London Math. Soc. 28 (1974), 683–699. (1974) Zbl0313.54044MR0362269
  4. Descriptive sets and the topology of nonseparable Banach spaces, Serdica Math. J. 27 (2001), 1–66. (2001) Zbl0982.46012MR1828793
  5. Descriptive topology, Recent Progress in General Topology, M. Hušek and J. van Mill (eds.), North-Holland, Amsterdam, London, New York, Tokyo, 1992, pp. 275–315. (1992) Zbl0805.54036MR1229129
  6. Čech analytic and almost K -descriptive spaces, Czech. Math. J. 43 (1993), 451–466. (1993) MR1249614
  7. Luzin theorems for scattered- K -analytic spaces and Borel measures on them, Atti Sem. Mat. Fis. Univ. Modena XLIV (1996), 395–413. (1996) MR1428772
  8. 10.1023/A:1013769030260, Czech. Math. J. 51 (2001), 791–818. (2001) MR1864043DOI10.1023/A:1013769030260
  9. On projections of nonseparable Souslin and Borel sets along separable spaces, Acta Univ. Carolin. Math. Phys. 42 (2001), 33–41. (2001) MR1900390
  10. Internal descriptions of absolute Borel classes, Topology Appl. 141 (2004), 87–104. (2004) MR2058682
  11. A converse of the Arsenin-Kunugui theorem on Borel sets with σ -compact sections, Fund. Math. 165 (2000), 191–202. (2000) MR1805424
  12. 10.1007/BF02392351, Acta Math. 149 (1982), 87–125. (1982) MR0674168DOI10.1007/BF02392351
  13. Borel-measurable selectors for compact-valued mappings in the non-separable case, Bull. Pol. Acad. Sci. Math. 23 (1975), 1043–1050. (1975) MR0410657
  14. Classical Descriptive Set Theory, Springer, New York etc., 1995. (1995) Zbl0819.04002MR1321597
  15. A remark on the uniformization in metric spaces, Acta Univ. Carolin. Math. Phys. 40 (1999), 65–74. (1999) MR1751542
  16. 10.4064/fm-58-2-149-157, Fund. Math. 58 (1966), 149–157. (1966) Zbl0143.07101MR0199339DOI10.4064/fm-58-2-149-157
  17. 10.1007/BF02394605, Acta Math. 120 (1968), 1–52. (1968) MR0237733DOI10.1007/BF02394605

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.