Countable network weight and multiplication of Borel sets

Fremlin, D.; Johnson, R.; Wajch, E.

doi:10.1090/S0002-9939-96-03488-0

Countable network weight and multiplication of Borel sets
HTML articles powered by AMS MathViewer

by D. H. Fremlin, R. A. Johnson and E. Wajch

Proc. Amer. Math. Soc. 124 (1996), 2897-2903

DOI: https://doi.org/10.1090/S0002-9939-96-03488-0

PDF | Request permission

Abstract:

A space $X$ Borel multiplies with a space $Y$ if each Borel set of $X\times Y$ is a member of the $\sigma$-algebra in $X\times Y$ generated by Borel rectangles. We show that a regular space $X$ Borel multiplies with every regular space if and only if $X$ has a countable network. We give an example of a Hausdorff space with a countable network which fails to Borel multiply with any non-separable metric space. In passing, we obtain a characterization of those spaces which Borel multiply with the space of countable ordinals, and an internal necessary and sufficient condition for $X$ to Borel multiply with every metric space.

References

K. Ciesielski and F. Galvin, Cylinder problem, Fund. Math. 127 (1987), no. 3, 171–176. MR 917142, DOI 10.4064/fm-127-3-171-176
Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
R. J. Gardner and W. F. Pfeffer, Borel measures, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 961–1043. MR 776641
Gary Gruenhage, Generalized metric spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 423–501. MR 776629
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
R. E. Hodel, On a theorem of Arhangel′skiĭ concerning Lindelöf $p$-spaces, Canadian J. Math. 27 (1975), 459–468. MR 375205, DOI 10.4153/CJM-1975-054-8
Roy A. Johnson, On product measures and Fubini’s theorem in locally compact space, Trans. Amer. Math. Soc. 123 (1966), 112–129. MR 197669, DOI 10.1090/S0002-9947-1966-0197669-0
Roy A. Johnson, Eliza Wajch, and Władysław Wilczyński, Metric spaces and multiplication of Borel sets, Rocky Mountain J. Math. 22 (1992), no. 4, 1341–1347. MR 1201097, DOI 10.1216/rmjm/1181072660
K. Kunen, Inaccessibility properties of cardinals, Ph.D. thesis, Stanford University, Palo Alto, 1968.
Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619
John C. Morgan II, Point set theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 131, Marcel Dekker, Inc., New York, 1990. MR 1026014
B. V. Rao, On discrete Borel spaces and projective sets, Bull. Amer. Math. Soc. 75 (1969), 614–617. MR 242684, DOI 10.1090/S0002-9904-1969-12264-0
B. V. Rao, On discrete Borel spaces, Acta Math. Acad. Sci. Hungar. 22 (1971/72), 197–198. MR 289312, DOI 10.1007/BF01896008
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
Michel Talagrand, Est-ce que $l^{\infty }$ est un espace mesurable?, Bull. Sci. Math. (2) 103 (1979), no. 3, 255–258 (French, with English summary). MR 539347