Poincaré-Volterra theorem

The set of Poincaré Volterra is a theorem of topology , one of the branches of mathematics . It is attributed to the two mathematicians Henri Poincaré and Vito Volterra , who formulated and proved it in first versions in the 1880s. The theorem deals with the question of the retransmission of topological properties through open continuous mappings and formulates a sufficient condition for this.

To the set of Poincaré Volterra there are a number of other versions and variations. The treatise by Peter Ullrich provides information about these and the history of the sentence .

Formulation of the sentence

In modern terms, the sentence is as follows:

Two Hausdorff rooms     and are given    . ${\displaystyle X}$ ${\displaystyle Y}$

  ${\displaystyle X}$   is contiguous and     is locally compact and locally connected and own a countable basis . ${\displaystyle Y}$

Furthermore, let it be     an open continuous mapping which satisfies the following additional conditions: ${\displaystyle {\phi }\colon \,X\to Y}$

Each element     has an open environment in   such a way that the restriction   with regard to the mutual subspace topologies represents a homeomorphism . ${\displaystyle x\in X}$   ${\displaystyle U_{x}\subseteq X}$   ${\displaystyle {\phi }{\upharpoonright }U_{x}\colon \,U_{x}\to {\phi }(U_{x})}$

Then:

${\displaystyle X}$   is also locally compact, locally connected and provided with a countable base.

Related result

The theory of Riemann surfaces has a result related to the above, which is also referred to in the relevant specialist literature as the Poincaré-Volterra theorem and which has proven to be an essential aid for the proof of Radó's theorem on Riemann surfaces .

This says the following:

Let a connected manifold     and a Hausdorff space be given    , which has a countable basis. ${\displaystyle X}$ ${\displaystyle Y}$

Furthermore, assume     a continuous mapping which satisfies the following additional conditions: ${\displaystyle {\phi }\colon \,X\to Y}$

For each element,     let the fiber overlying the element be   a discrete subspace of    . ${\displaystyle y\in Y}$   ${\displaystyle {\phi }^{-1}(y)}$ ${\displaystyle X}$

Then:

Also     has a countable base. ${\displaystyle X}$

literature

• Heinrich Behnke , Friedrich Sommer : Theory of the analytical functions of a complex variable (=  The basic teachings of the mathematical sciences in individual representations . Volume 77 ). Springer Verlag, Berlin (among others) 1965.
• Nicolas Bourbaki : General Topology (=  Elements of Mathematics . Part I). Addison-Wesley Publishing (et al.), Reading, Massachusetts (et al.) 1966 ( MR0205210 ).
• Otto Forster : Riemann surfaces (=  Heidelberg pocket books . Volume 184 ). Springer Verlag, Berlin / Heidelberg / New York 1997, ISBN 3-540-08034-1 ( MR1472025 ).
• Krzysztof Maurin : The Riemann Legacy: Riemannian Ideas in Mathematics and Physics (=  Mathematics and its Applications . Volume 417 ). Kluwer Academic Publishers, Dordrecht ( inter alia ) 1997, ISBN 0-7923-4636-X ( MR1472025 ).
• Horst Schubert : Topology. An introduction (=  mathematical guidelines ). 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
• Peter Ullrich: The Poincaré-Volterra Theorem: From Hyperelliptic Integrals to Manifolds with Countable Topology . In: Arch. Hist. Exact Sci . tape 54 , 2000, pp. 375-402 ., Doi : 10.1007 / PL00021243 . MR1741400

References and comments

1. ^ Ullrich: The Poincaré-Volterra Theorem: From Hyperelliptic Integrals to Manifolds with Countable Topology. In: Arch. Hist. Exact Sci . tape 54 , 2000, pp. 375 ff .
2. ↑ Bourbaki: pp. 114-116.
3. ↑ A continuous mapping of this kind is called in the topology also locally topologically ; see. Schubert: p. 216.
4. ↑ Maurin: pp. 453-454.
5. ↑ Behnke-Sommer: pp. 453–454.
6. ↑ Forster: pp. 165–166.
7. ↑ A mapping between two topological spaces that satisfies this additional condition is also called a discrete mapping ; see. Forster: p. 18. In topology, discrete mappings occur primarily in connection with overlays . Because here every overlay image is discrete ; see. Schubert: p. 216.