The theory of Riemann surfaces has a geometric and an analytic part. The former deals with the axiomatic definition of a Riemann surface, methods of. Riemann Surfaces. Front Cover. Lars V. Ahlfors, Leo Sario. Princeton University Press, Jan 1, – Mathematics – pages. A detailed exposition, and proofs, can be found in Ahlfors-Sario [], Forster Riemann Surface Meromorphic Function Elliptic Curve Complex Manifold.

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For instance, it cannot be proved by analytical means that every surface which satisfies the axiom of countability can be made into a Riemann surface. Certain characteristic properties which surfacees or may not be present in a topological space are very important not only in the’ general theory, but in particular for the study of surfaces. It so happens that ahlors superficial knowledge is adequate for most applications to the theory of Riemann surfaces, and our presentation is influenced by this fact.

## Lars V. Ahlfors, L. Sario-Riemann Surfaces

At the saio end of the chapter it is then shown, by essential use of the Jordan curve theorem, that every surface which satisfies the second axiom of countability permits a.

A Migllborlwod of a set A c 8 is a set V c 8 which. Such a basis is a system fJI of subsets of 8 which satisfies condition B The intersection of any finite collldion of sets in fJI is a union of sets in The main demerit of this approach is that it does not yield complete results. The chapter closes with the construction of a triangulation. The compomntB of a locally connuted space are Bimtdtamously open and cloBed.

If 01 is not empty it meets at least one P. In other instances the analytical method becomes so involved that it no longer possesses the merit of elegance. We call this topology on S’ the relative topology induced by the topology on B. The fundamental group is introduced, and the notion of bordE. The following conditions shall be fulfilled:. The sum of two topological spaces 81, Sa is their union 81 u Bz on which the open seta are those whose intersections with 81 and Sa are both open.

Suppose that p belongs to the component 0, and let V p be a connected neighborhood. The space obtained by identification can be referred to as the qvotientspau of S with respect to the equivalence relation whose equivalence classes are the sets P. We shall say that a family of closed seta has the finite inter- M. C C X Xcomo For instance, a compact space can thus be covered by a finite number of from an arbitrary basis.

A space iB compact if and only if every open covering contains a finite BUbcovering. In other words, p belongs to the boundary of P if and only if every V p interaecta Pas well as the complement of P.

## Lars V. Ahlfors, L. Sario – Riemann Surfaces

The requirements are interpreted to hold also for the empty collection. As a consequence of 3B every point in a topological space belongs to a maximal connected subset, namely the union of all connected sets which contnin the given point. This definition has an obvious generalization to the case of an arbi- trary collection of topological spaces. For the points of the plane Jl2 we shall frequently use the complex notation The sphere 81, also referred to as the u: It exists, for it can be obtained as t lw intersection of all closed sets which contain P.

It shows that every open set in a locally connected space is a union of disjoint regions.

### Lars V. Ahlfors, L. Sario-Riemann Surfaces – PDF Drive

The definition applies also to subsets in their relative topology, and we can hence apeak of connected and disconnected subsets.

In most a third requirement is added: Every open subset of a locally connected space is itself locally connected, for it has a basis conaisting of part of the basis for the whole space.

AI The union of any collldion of open sets is open. In the case of a aubaet it is convenient to replace the relatively open seta of a covering by corresponding open seta of the whole apace.

For complete results this derivation must be based on the method of triangulation. In most a third requirement is added:.

Tbia is to be contrasted with the formula AnBcAnB which is weaker inasmuch as it gives only an inclusion. This shows that 0 is open. Surface acetylation of bacterial cellulose Surface acetylation of bacterial cellulose. A topology 9″ 1 is said to be weaker than the rimeann r 2 if r 1 c r ‘1.

### Springer : Review: Lars V. Ahlfors and Leo Sario, Riemann surfaces

The following theorem is thus merely a rephrasing of the definition. There is a great temptation to bypass the finer deta. Sario – Riemann Surfaces Alexandre row Enviado por: From the connectedness of P. Surface nano-architecture of a metalorganic framework Surface nano-architecture of a metalorganic framework.

The boundary of P is formed by all points which belong neither to the interior nor to the exterior.