an:01423834
Zbl 0967.01007
Ullrich, Peter
The Poincar??-Volterra theorem: From hyperelliptic integrals to manifolds with countable topology
EN
Arch. Hist. Exact Sci. 54, No. 5, 375-402 (1999).
00062533
1999
j
01A55 30-03 54-03
Poincar??-Volterra; topology; set theory
The Poincar??-Volterra theorem states that a multi-valued analytic function on \(\mathbb C\) has at most countable values \(f(z)\) for any fixed \(z\), or said otherwise, the fibers of a Riemann surface are finite or countable. The paper describes the birth of this theorem, and the various claims by Cantor, Weierstra??, Vivanti, Poincar?? and Volterra in the 1870's and 1880's on this topic. Precisely written, this lively paper recounts what turned out to be a decisive moment in the emergence of set-theoretic ideas in the theory of functions of one complex variable, which contributed, as consequence, to their acceptance in main-stream mathematics by the end of the 19th century.
Martin Andler (Versailles)