Baire sections for group homomorphisms

Graf, S.; Mägerl, G.

doi:10.1090/S0002-9939-1982-0660616-3

Baire sections for group homomorphisms
HTML articles powered by AMS MathViewer

by S. Graf and G. Mägerl

Proc. Amer. Math. Soc. 85 (1982), 615-618

DOI: https://doi.org/10.1090/S0002-9939-1982-0660616-3

PDF | Request permission

Abstract:

The following result is proved: Let $X$ and $Y$ be compact topological groups and $p$ be a continuous group homomorphism from $Y$ onto $X$. Then there exists a map $q$ from $X$ to $Y$ such that $p \circ q = {\text {i}}{{\text {d}}_X}$ and ${q^{ - 1}}(B)$ is a Baire set in $Y$ for every Baire subset $B$ of $X$.

References

M. Bockstein, Un théorème de séparabilité pour les produits topologiques, Fund. Math. 35 (1948), 242–246 (French). MR 27503, DOI 10.4064/fm-35-1-242-246
Siegfried Graf, A measurable selection theorem for compact-valued maps, Manuscripta Math. 27 (1979), no. 4, 341–352. MR 534799, DOI 10.1007/BF01507290
P. J. Higgins, Introduction to topological groups, London Mathematical Society Lecture Note Series, No. 15, Cambridge University Press, London-New York, 1974. MR 0360908
Joseph Kupka, Strong liftings with application to measurable cross sections in locally compact groups, Israel J. Math. 44 (1983), no. 3, 243–261. MR 693662, DOI 10.1007/BF02760974
K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 397–403 (English, with Russian summary). MR 188994
A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissertationes Math. (Rozprawy Mat.) 58 (1968), 92. MR 227751
Marc A. Rieffel, On extensions of locally compact groups, Amer. J. Math. 88 (1966), 871–880. MR 202912, DOI 10.2307/2373085